h r t l C p t r S n ELSEVIER eo e ica Com u e Scie ce 1 (1995 3-5
Mellin transforms and asymptotics: Harmonic sums
Philippe Flajolet*, Xavier Gourdon, Philippe Dumas INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France
e ica e o o K u a G e ui o ei io ee i g wo ks o Me i a s o ms a com i a o ics
Abstract
is su ey ese s a u i ie a esse ia y se -co ai e a oac o e asym o ic a a ysis o a a ge c ass o sums a a ise i com i a o ia ma ema ics isc e e o a i is ic mo e s a e a e age-case a a ysis o a go i ms I e ies o e Me i a s o m a c ose e a i e o e i eg a a s o ms o a ace a ou ie e me o a ies o a mo ic sums a a e su e osi io s o a e a i a y " a mo ics" o a commo ase u c io I s i ci e is a ecise co es o e ce e wee i i i ua e ms i e asym o ic e a sio o a o igi a u c io a si gu a i ies o e a s o me u c io e mai a ica io s a e i e a ea o igi a a a s uc u es o a i is ic a go i ms a commu ica io eo y
ie eo ie e e i oke u k io e u I eg a e 1s em ce a es Ge ie we c es ma c e a e e Ge ie e e A a ysis mi ei a e e i e
a ma Me i
Introduction
Hjalmar Mellin (1854-1933, see [59] for a summary of his works) gave his name to the MeIlin transform that associates to a function f (x) defined over the positive reals the complex function f * (s) where
f * (s) = f(s1 dx. (1)
The change of variables x = e """ shows that the Mellin transform is closely related to the Laplace transform and the Fourier transform. However, despite this connection,
Co es o i g au o Emai i i e a o e i ia
E se ie Scie ce SSD1 3 -3975(95 -
4 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58
there are numerous applications where it proves convenient to operate directly with the Mellin form (1) rather than the Laplace-Fourier version. This is often the case in complex function theory (asymptotics of Gamma-related functions [64, 65]), in num- ber theory (coefficients of Dirichlet series, after Riemann), in applied mathematics (asymptotic estimation of integral forms), and in the analysis of algorithms (harmonic sums introduced below). Thus, throughout this paper, we operate directly with the Mellin transform. The major use of the Mellin transform examined here is for the asymptotic analysis of sums obeying the general pattern
G(x) = E g(pk x), (2)
either as x 0 or as x -+ co. Following a proposal of [31], sums of this type are called harmonic sums as they represent a linear superposition of "harmonics" of a single base function g(x). Harmonic sums surface at many places in combinatorial mathematics as well as in the analysis of algorithms and data structures. De Bruijn and Knuth are responsible in an essential way for introducing the Mellin transform in this range of problems, as attested by Knuth's account in [56, p. 131] and the classic paper [16] which have been the basis of many later combinatorial applications. For instance, the analyses of the radix-exchange sort algorithm [56] and of the expected height of plane (Catalan) trees [16] involve the quantities 1 n n ( 2nk) S„ = E [i - (i - - and T„ = E d(k) (3) k= k = ( where d(k) is the number of divisors of k and (g is the binomial coefficient. These discrete quantities have continuous analogues that are harmonic sums
_ k] G(x) E i and H(x) = E (k e - k (4) k= k= and elementary arguments establish that asymptotically S„ G(n) and T„ H (n). There are inherent difficulties in the asymptotic analysis of the discrete sums S„, 7'„ or their continuous analogues G(x), H (x). The divisor function in T„ and H (x) fluctuates heavily in a rather irregular manner (for instance it equals 2 if and only if k is prime). The quantities S„ and G(x) look more innocuous; however, a Mellin based analysis to be shown later reveals that they involve subtle fluctuations of a tiny amplitude, less than 10 -5 . Such behaviors preclude the use of elementary real asymptotic techniques in most sums of this type. Over the past 20 years, perhaps some 50 odd analyses of algorithms or related evaluations of parameters of combinatorial structures have involved in a crucial way a Mellin treatment of harmonic sums of sorts. We shall organize and detail in later sections several of these examples that can be broadly categorized, in terms of their range of applications, as follows. Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-5 5
— igi a sea c i g me o s a i -e c a ge so igi a ees igi a sea c ees [5 ] a ei ge e a i a io s ike " ucke " ees [3 ] a icia ees [5 ] su i ees [ ] igi a ees a ia ce a iase - i mo e s [5 79] ee ma i u a io a go i ms eig a s ack e [1 ] egis e a oca io [5 ] S i g sea c i g a e occu e ce o a e s i s i gs [ ] (a e e e ces e ei wi a ica io s o ca y o aga io i i a y a e s [57] a a a com essio [ 7] Mu i ime sio a sea c i g o ems [ 9] a Euc i ea ma c i g [7 ] E e i e as i g a g i i e me o s [7 73] a a e a is i u e a go i ms ike e ea e e ec io ec ique o [7 ] commu ica io o oco s ase o e Ca e a akis— sy ako sc eme [ 39 3] a a e so i g e wo ks ase o a c e s o —e e sc eme [75] i i e-a -co que a go i ms me geso [ 5] a geome ic ma ima i i g [ ] a omi e a a s uc u es ike ski is s [5 ] o a i is ic es ima io a go i ms o a i is ic cou i g [ 7] a a i e sam i g [ 3] o a o ima e cou i g [ 71] I ma y cases o a y igi a sea c ees a mo ic sums a e a a e a i e o e me o o " ice i eg a s" a is iscusse i e ai e sew e e [3 ] Some ie accou s o e me o a e gi e i e ooks y o i [ ] Kem [51 1 1] Ma mou [ ] a i e a ook c a e [ ] We o ow e e e a c i ec u e o e i o ma su ey [31] Ge e a o e ies o e Me i a s o m a e usua y ea e i e ai i ooks o i eg a a s o ms ike ose o oe sc [1 ] Wi e [ ] o i c ma s [ ] Asym o ic me o s i co ec io wi Me i a s o ms a e iscusse wi i e co e o a ie ma ema ics i ea ises y a ies [13] i g e [17] a Wo g [ ] I a icu a ou wo k is c ose i s i i o Wo g s w o iscusses e e si e y a a a ogue o a mo ic sums i e o m o " a mo ic i eg a s"
G( = 3 1(1c g(i 1(k (5 o (I is co i uous case y a c a ge o a ia es o e may a ways assume a k (K K Some o e ma y uses i um e eo y a e ea e o i s a ce i [ 7 ]
Principles. A ma o use o Me i a s o m i asym o ic a a ysis is o es ima i g asym o ica y a mo ic sums ( e 1 1 a e e "am i u es" e / k a e e " eque cies" a g( is e " ase u c io " a mo ic sums e uce o usua ou ie se ies w e e ase u c io g( is ake o e a com e e o e ia g( = e a e eque cies a e e i ege s a --E- k 1 Me i a s o ms a e "se a a io " o e y e Me i a s o m as e i e i (1 co e ges i a s i o e com e a e ca e e u ame a s i y 6 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58
a direct change of variables, the Mellin transform of g (px) is p times the transform g*(s) of g(x). Thus, by linearity, the Mellin transform of a general harmonic sum is (conditionally) G* (s) = A (s)• g* (s), (6) where
A(s)= E1kk5 g* (s g (x) dx. It therefore factors as the product of the transform of the base function and of a generalized Dirichlet series: Mellin transforms applied to harmonic sums "separate" the amplitude—frequency pair from the base function. The inversion theorem for Mellin transforms is analogous to Fourier inversion, 1 f (x) = 2in— c_ f * (s) x' ds . (7)
There, the integration line 9i (s) = c should be taken in the fundamental strip of the Mellin transform. Basic functional properties of the Mellin transform are recalled in Section 1 (Theorem 1) and the separation property is expressed by Lemma 2 of Section 3. 2. Poles of transforms and asymptotics of originals. There is a fundamental corres- pondence between the asymptotic expansion of a function (either at 0 or co) and singularities of the transformed function. First and foremost, Mellin asymptotics crucially relies on analytic continuation of transforms. Consider the problem of asymptotically expanding f(x) as x 0 when f * (s) is known to be meromorphically continuable in C. The starting point is the inversion formula (7). The line of integration is then shifted to the left while taking residues into account. For instance when c > 0 and f * (s) has simple poles at the nonpositive integers, each simple pole at s = — m contributes a term proportional to et since Res ( f * (s) x"), _„, = Res( f * (s)),„ _ ,„ • xm (There Res(g(s)),--,„ denotes the residues of g(s) at s = so .) Thus, globally integrating along an infinite rectangle with sides 9i (s) = c and 91(s) = — M — 1/2 (M an integer) gives by the residue theorem
f (x) = E (Res f * (s)),, _ „, xin + 0 (x m + / ). (8) m=o Some additional decay condition on f*(s) is evidently necessary in order to justify (8). The computation outlined above reflects a general phenomenon: Poles of a Mellin transform are in direct correspondence with terms in the asymptotic expansion of the original function at either 0 or + cc. Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995) 3-58 7
o e asym o ic e a ua io o a a mo ic sum G(x) is i ci e a ies o G* (s) o i e e i ic e se ies A (s a e a s o m g* (s) a e eac a a y ica y co i ua e a o co o e g ow is assum io is ea is ic o ma y a u a y occu i g ase u c io s a o am i u e a eque cies gi e y " aws" a a e egu a e oug e u ame a co es o e ce is e o e i Sec io ( eo ems 3 a 4); i s a ica io o a mo ic sums is s e e ou i Sec io 3
Example. e mo e o o e a io o Me i asym o ics is we i us a e y e a mo ic sum
G (x) = E - 1 k ( og k) e k (9 k = 1 I is case e c assica equa io s 1 F(s) e s a C(s E O k= 1 e i e e Gamma u c io o Eu e a e iema e a u c io [ 5] us F(s) is e Me i a s o m o e ase u c io g( = e a a sim e com u a io yie s
s) A - = 111 -(1 - 2 1 ')C(s). ( s I is way e a s o m o G( is ou o e s = [ 1- G ( s i ( og (2s) + (1 - 1 5 C (2s)] • F (s). (1 A a y ica y e e ese a io (1 is easi y us i ie w e 9i (s) > 1, a co i io a e su es simu a eous y a so u e co e ge ce o e i ic e se ies a o e Me i i eg a I is k ow a e Gamma u c io a e e a u c io a e o co i ua e o e com e a e F (s) as sim e o es a e o osi i e i ege s w i e C(s as o y a sim e o e a s = 1. A so F (s) ec eases as a o g e ica i es w i e (s is o y o mo e a e g ow us g o a y G* (s) ge s sma o s + ioo a e i eg a io co ou ca egi ima e y e s i e o e e o i s a ce y swee i g e i eg a- io co ou i e e ica i e 91(s = - 5/ o e ge s om e o es a s = - 1, -
G( = og - + c i + c + ( 51
wi c 1 = 7 ( - a c = - C ( - e i s ew e ms a e e e e mi e i a ma e o seco s wi e e o a com u e a ge a sys em ike Ma e a "k ows" e e a sio s o (s a F (s) a oi s o i e es ike C( = - C( = - og / 7c
8 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995) 3-58
The computation is easily carried out to any order, and one finds _ G(x) — log (2,2k_ oc(-20xk. k=1 k Such an expansion might be obtained otherwise via a reduction to Euler—Maclaurin summation but application of this formula would be hampered by the alternation of sign and by the presence of the logarithmic factor. Mellin asymptotics is the approach of choice here.
The method is susceptible to a large number of variations. It is applicable each time the Dirichlet series of an amplitude—frequency pair is reducible to a standard function, like for instance
CO 1 1 (k 2 E = c (s). k =0 (2kY = 1 — 2 - s ' k = 1 In this way sums involving geometrically increasing frequencies or highly fluctuating amplitudes of an arithmetic nature can be analyzed effortlessly, see Sections 4-7. Another feature is that the correspondence between poles of transforms, and asymp- totic terms of originals fares both ways; this fact allows for base functions that admit of no explicit transform, like
g(x) = - 2 as well as exotic Dirichlet series like
og k ( — 1) k A() = E k=1 /1 k ks This flexibility explains the power of the Mellin method in asymptotic analysis, which would apply for instance to a harmonic sum like
G(x) = E (_ 1)k (log k)e -k x k=1 The general methodology for analyzing such "implicit" sums is discussed in Section 8. Plan of the paper: Part I is devoted to the general functional and asymptotic properties of Mellin transforms. Our presentation assumes only rudiments of complex analysis (contour integrals, residue theorem) as found for instance in [81, 85]. The general framework is conveniently built on the Lebesgue integral [81, Ch. X] allowing dominated convergence properties, rather than on the Riemann integral. This distinc- tion is however somewhat immaterial as specific examples can be dealt with using Riemann integrals only. A number of examples related to combinatorial analysis and the analysis of algorithms are then presented in Part II. Finally, methods for dealing with wider classes of sums form the subject of Part III. Ph. Flajolet et al.1 Theoretical Computer Science 144 (1995) 3-58 9
PART I. MELLIN TRANSFORMS AND ASYMPTOTICS
In this part, we lay down the basic framework of Mellin asymptotics. Section 1 gives the basic functional properties of Mellin transforms and Section 2 discusses the fundamental correspondence between asymptotic properties of an original function and singularities of its transform. Section 3 develops the basic treatment of harmonic sums that form the subject of this paper. Section 4 briefly examines consequences for general summatory formulae.
1. Basic properties
We start by recalling the salient properties of the Mellin transform. The definition domain of a Mellin transform turns out to be a strip. We thus introduce the notation 1 for the o e s i of complex numbers s = a + i such that a a
Definition 1. Let ( be a locally Lebesgue integrable over (0, + co). The Me i a s o m of ( is defined by +co [ ( ; 3] (s = 1 ( 5- dx (11)
The largest open strip a 13 in which the integral converges is called the u ame a s i
From the decomposition 1 ' =1 1 + , we see that a is the infimum of all A such that ( A is integrable over (0, 1] and fl is the supremum of all such that (x x is integrable over [1, + co). Most functions have an order at 0 and co so that an existence strip for (s can be guaranteed.
Lemma 1. e co i io s
( = ( u ( = ( -o+ --• +
w e u gua a ee a (s e is s i e s i — u —
Thus, existence of a Mellin transform is granted for locally integrable functions' whose exponent in the order at 0 is strictly larger than the exponent of the order at infinity. e asym o ic o m o ( a 0 co s ai s e e mos ou a y o e u ame a s i o (s ; e asym o ic o m a + oo co s ai s e ig mos
I e seque a u c io s co si e e a e aci y assume o e oca y i eg a e 10 Ph. Flajolet et al.1 Theoretical Computer Science 144 ( 1995) 3-58 boundary. Monomials x`, including constants, thus do not have transforms under Definition 1. As the integral defining f *(s) depends analytically on the complex parameter s a Mellin transform is in addition analytic in its fundamental strip. For instance, the function f (x) = (1 + x) -1 is 0(x° ) at 0 and 0(x -1 ) at infinity, hence a guaranteed existence strip for f * (s) that is <0, 1>, which here coincides with the fundamental strip. In this case, the Mellin transform may be found from the classical Beta integral [85, p. 254] to be
f (s) = si i s which is analytic in <0, I>, as predicted. The function g(x) = e satisfies
e x - 1, e - x = 0(x') for any b > 0 x-o* so that its transform, known as the Gamma function [85, Ch. XII] +0,, F (s) = i e'xs--1 dx, (12) J o is a priori defined in <0, + co> and analytic there. Let ( be the (Heaviside-like) step function defined by H(x) = 1 if 0 x <1, H(x) = 0 if 1< x. (13) Then, 1 H* (s) = - , s E + co>. s
The complementary function 171(x) = 1 - H(x) satisfies H* (s) = - s 1 for s e <- co, 0>, where the fundamental strips corresponding to H and ii are disjoint.
Functional properties. Simple changes of variables in the definition of Mellin transforms yield a basic set of transformation rules summarized in Fig. 1 and Theorem 1 below.
Theorem 1 (Functional properties). Let f (x) be a function whose transform admits the fundamental strip [f (Px); = tcss f*(s), s fl> EEk Akf(mkx); = (Ek„ Ak • f * (s), itc finite, > O. di [x' f (x); s] f * (s + v), s E a i Ef (x°); s] = —1 f* S E ( * (s 3 xv f (x) * (s + i k( s ( log x —f * (S ds a 13 5 ( — s * (s a e = dx —f ( — (s — 1 * (s — 1 Fig. 1. Basic functional properties of Mellin transforms. The table lists the original function, its Mellin transform, and the validity strips. The most important rule is the rescaling rule that gives the transform of (/ux) as s (s via the change of variables x 1—* fix, provided that it > 0. By linearity of the transform, one also has A Ji[E i(1,1kx); si=( E—) .f* (s k whenever k ranges over a finite set of indices. This formula can usually be extended to infinite sums that define the harmonic sums already mentioned in the introduction, see Lemma 1 and Theorem 5 below. For instance from e' ( = = e - e- e-3 1 — ex one finds 1 1 1 (s = F (s)• + • •• = F(s)C(s), as long as 91(s) > 1, a condition that simultaneously entails absolute convergence of the Mellin integral of ( and of the sum defining the zeta function. 12 Ph. Flajolet et al. 1 Theoretical Computer Science 144 ( 1995 ) 3-58 A particular case of the rule for powers is [ (x-1 ); s = — (— s Since the function 1/x exchanges 0 and oo, this permits to limit consideration of asymptotic properties of a function to one of the two places 0, + oo. A useful application of the rule is for the transform of the Gaussian function, [e - ; s] = (-s 2 2 which arises in estimates of sums involving binomial coefficients. The formal rule (s = ( (log x) dx ds is readily justified analytically by "differentiation under the integral sign". (As com- plex-differentiable implies analytic, this also supports our earlier claim that Mellin transforms are analytic in the fundamental strip.) Thus [ ( (log x); s] = — s (s For instance, the transform of ( log x is - 1/s , the transform of e log x is (s Conversely, the rule for transforming the derivative of an original function is best enunciated in terms of the operator = x — . dx Frequently the fundamental strips of ( and O ( have a nonempty intersection a 3 This is ensured in the common case where ( = 0(x) as x 0 and ( 0(x - ° as -■ co w e e e e ( satisfies the same estimates (i.e., the asymptotic form of the derivative of coincides with the derivative of the asymptotic form of Under this sufficient condition, with ( continuous and piecewise differentiable, integration by parts yields + + ef ( dx = [ ( s];" - s ( dx, a 91(s 13 a e e m [ ( s E equals 0 for s e a 1 us [ef (x) ; = - s (s This permits to deduce Mellin transforms of primitive functions by identification. For instance, from the transform of (1 + x) - 1 , we find I di [log (1 + x), s] = — 1 9( 1 s sin i s Ph. Flajolet et al. I Theoretical Computer Science 144 (1995) 3-58 13 e F (s + co e — 1 F (s) < — 1, e - — 1 + F (s — —1 - 2 IF(1 s + co 1 C 1 + si i s it og(1 + — 0> s si ICS 1 ( 1 1 + co (-1 q ( og ( —a + co Ice (s + Fig. 2. A microdictionary of Mellin transforms. I e same ei (s + 1 [e - — 1 s] = — F (s — 1 91(s a so o see ig o e o a s o ms o a a y ic u c io s A oug we o o make use o i e e e e is a ui u a oac o e e e mi a io o a s o ms o a a y ic u c io s a is ase o a c assica oo i eg a e ese a io ue o a ke [ 3 5] A a ke co ou e is a sim e oo a s a s i e u e a - a e ea + co ci c es a ou e o igi cou e c ockwise a e u s o + co i e owe a - a e I ( is a a y ic i some o e se a co ai s [ + co a sa is ies easo a e g ow co i io s e a ke s o mu a o s — 1 [ ( s] = (w ( _w)1 0< a < (1 i si i s o e oo is ase o s i ki g e co ou owa s e ea a is a usi g e ac a i e i i i esima imi e i eg a is ( S 1 mu i ie y (e -i" — e i" = — i si i s a ke s o mu a e mi s o e e mi e a a s o ms o a io a u c io s as we as ma y a s o ms o me omo ic u c io s See [77 3 5] Inv r n Wi agai s = a + i a e c a ge o a ia es = e - Y e Me i a s o m ecomes a ou ie a s o m (s =f ( 5 - 1 = — - - 3 (e - Y e - "e -i Y y - 14 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995) 3-58 This links the two transforms; hence it provides directly inversion theorems for Mellin transforms. We cite here two main versions, one related to Lebesgue integration, the other to Riemann integration. Theorem 2 (Inversion). (i) Let f(x) be integrable with fundamental strip Proof. See [18, 82, 86]. C7 Fig. 2 presents a few classical original-transform pairs ( f, f *). More can be found in standard tables of integral transforms [19, 66, 67] and in [61, 77]. 2. The fundamental correspondence There is a very precise correspondence between the asymptotic expansion of a function at 0 (and co), and poles of its Mellin transform in a left (resp. right) half-plane. Each individual term in such an asymptotic expansion of f(x) having the form xe (log x)k is associated to a pole of f *(s) at s = — c with multiplicity k + 1. The correspondence fares both ways. It is thus the basis of an asymptotic process: For estimating asymptotically a function F(x) - typically a harmonic sum -, determine its Mellin transform and translate back its singularities into asymptotic terms in the expansion of F(x). Let (s) be meromorphic at s = so : it admits near so a Laurent expansion k 0(s) = E ck ( s — so (15) k — The function Os) has a pole of order r if r > 0 and c _ 0 0, it is analytic (regular, holomorphic) at so if r = 0. A singular element of 0(s) at so is an initial sum of the Laurent expansion (15) truncated at terms of order 0(1) or smaller. Definition 2 (Singular expansion). Let 0(s) be meromorphic in Q with 9' including all the poles of 0(s) in Q. A singular expansion of 0(s) in Q is a formal sum of singular elements of 0(s) at all points of 99. h l j l t t l I h r t l C p t r S n 44 ( - 8 15 When E is a singular expansion of Cs), in Q, we write E (s e 0). For instance, one has 1 + 2 + 3(s + 1)1 s (s + 1) Ls + 1 s= -1 1 1 1 + — —1 + (s e < — 2, + 2> ), (16) s s L s = 1 where the point of expansion may be indicated whenever needed as a subscript to the corresponding singular element. The expansion (16) is a concise way of combining information contained in the Laurent expansions of the function 0(s) s - (s + 1)_ 1 at the three points of 99 = { — 1, 0, 1}: s- 1 1_ 1 + = (S + ) 1 + 2 + 3(s + 1) + •-• , = s-■ -1 = - (S - 1 + 7 (S - 1 + • • • s- 1 Example 1. The Gamma function. The Mellin transform of the function e - x defines the Gamma function, F(s) = e - x x 3-1 dx, for 91(s) > 0. Integration by part permits to verify the well-known functional equation F(s + 1) = sT(s), which allow to extend F (s) to a meromorphic function in the whole of C. The function (s) satisfies F(1) = 1, and from the functional equation one has T (s + m + 1) T (s) = s(s + 1)(s + 2) • • -(s + m) Thus, F(s) has poles at the points s = — m with m E NI, near which (-1' 1 F (s) m! s + m' so that the Gamma function admits ( — 1)k 1 F E (s e C) (17) k= k! s + k as singular expansion in the whole of C. Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 (s Order at 0: 0(xa) Leftmost boundary of f.s. at 9i(s) = —a Order at + co : 0(xb ) Rightmost boundary of f.s. at at 9i(s) — Expansion till 0(xv) at 0 Meromorphic continuation till 9i(s) — y Expansion till 0(x 5) at co Meromorphic continuation till 91(s) —(5 ( —1 k k Term xa (log x)k at 0 Pole with singular element (s + +1 ( —1)k k Term X' (log x)k at co Pole with singular element (s + a k +1 ig 3 e u ame a co es o e ce as ec s o e i ec ma i g ( eo em 3 Direct mapping. The function e - x has a Taylor expansion at x = 0: — k e - = E (18) k= 0 k • There is a striking coincidence of coefficients in the Taylor expansion (18) of the original function e - x and in the singular expansion (17) of the transform (s expressed by the rule 1 1— s + k • This is in fact a completely general phenomenon. Theorem 3 (Direct mapping; see Fig. 3). e ( a e a a s o m (s wi o em y u ame a s i a (i Assume a ( a mi s as 0 + a i i e asym o ic e a sio o e o m ( = E c k ( og k + 0 (x 7 ) (19) (4, k e A w e e e sa is y — y < — c a e k a e o ega i e e (s is co i u- a e o a me omo ic u c io i e s i —y i w e e i a mi s e si gu a e a sio ( — 1 k k f ( - E (s ok+ 1 (s ( k A e (ii) Simi a y assume a ( a mi s as + co a i i e asym o ic e a sio o e o m (19 w e e ow 13 — — y. e (s is co i ua e o a me omo ic Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 17 function in the strip ( — 1 k k (s — E ck _ _ )k+1 (s e Thus terms in the asymptotic expansion of f (x) at 0 induce poles of (s) in a strip to the left of the fundamental strip; terms in the expansion at + co induce poles in a strip to the right. Proof (see [18]). Since II( f (1 / x), s) = — ( f (x), — s), it suffices to treat the case (i) corresponding to x — 0. By assumption, the function g(x), g(x) = f (x) — E C4 k x (log x)k , (4,k)eA satisfies g(x) = O(x). For s in the fundamental strip, a split of the definition domains yields 1 f * (s) = f (x) xs dx + (x) sdx o .1 1 1 = g (x) X 1 + E C4 k ( og x)x x' dx o (4,k)e A + co f (x) xs dx (20) In the last line of (20), the first integral defines an analytic function of s in the strip < —y, + oo > since g(x) = 0 (xv) as x 0; the third integral is analytic in < — co fl>, so that the sum of these two is analytic in < — y, /3>. Finally, straight integration expresses the middle integral as ( — 1 k k E cA k (s k+1 (4,k)e A which is meromorphic in all C and provides the singular expansion of f * (s) in the extended strip. In the case where there exists a complete expansion of f (x) at 0 (or co), the transform f * (s) becomes meromorphic in a complete left (or right) half-plane. This situation always occurs for functions that are analytic at 0 (or + cc Example 2. The zeta function. We gave earlier the Mellin pair e' f (x) = f * (s) = F (s) (s) 1— e ' 18 Ph. Flajolet et al.1 Theoretical Computer Science 144 (1995) 3-58 with fundamental strip < co>. The function f(x) is exponentially small at infinity and it admits a complete expansion near x = 0, CO k e = E k+ 1 ii e k= —1 (k + 1 which defines the Bernoulli numbers k == 1, B 1 = 1, etc. Thus, (s = (s)C(s) is meromorphic in the whole of C with singular expansion F (s) (s) E k+1 1 (21) k = _ (k + 1)! s + Therefore, ((s) is meromorphic in the whole of C (this is often proved by means of its functional equation). In addition comparison of the singular expansions of F(s) in (17) and r(s)(s) in (21) yields 1 — CO) = 1 C( ni) = m s — 1 — 1m + 1 Ex pl The transform of (1 + x) -1 . The function f(x) (1 + x) -1- has <0, 1> as its fundamental strip. The two expansions, 1 CO 1 _ E ( o ( o+ a = + + „ 1+ n -1 translate into " f E —1 (s < —co , I.>) and n=0 s + n ( 1)" f — E (sE +co s — n which is consistent with the known form, — f * (s) = E (se C). sin i s „ S n The next example illustrates the fact the Mellin analysis may be applied to functions without explicit transforms. Ex pl 4 A nonexplicit transform. The function f (x) = (cosh (x)) -1 / is exponenti- ally small at co and near 0 it satisfies 1 1 7 139 5473 = 1 + + + ( 1 cosh (x) 4 96 5760 645120 Ph. Flajolet et al. 1 Theoretical Computer Science 1 ( 1995 ) 3-58 19 so a i s a s o m (s is me omo ic i C a 1 1 7 1 139 1 5473 1 + + + s s + 9 s + 57 s + 51 s + A ge e a i ci e a so e i es om e oo o eo em 3 subtracting from a function a truncated form of its asymptotic expansion at either or co does not alter its Mellin transform and only shifts the fundamental strip. A i s a ce is o i e y e equa i ies s = F (s), S E + CO (e- —1 s) = F (s), s e — 1 ( e ious y es a is i g usi g i eg a io y a s a s eci ic o e ies o e e o e ia e o owi g oo o ( emo s a es e ge e a ec ique o is a icu a e am e ake e u c io 1 F* (s) = - + I (e — 1 s -1 + I e s s i Co si e a io o o i eg a s s ows a e u c io is me omo ic i —1 + co I s es ic io o + co is F(s)= e - xxs a i s es ic io o —1 is (e - — 1)xs -1 1 ° is a gume s ows a e a s o ms o e a o e — 1 a e e eme s o e same me omo ic u c io i i e e s i s Converse mapping. U e a se o mi co i io s a co e se o e i ec Ma i g eo em a so o s e si gu a i ies o a Me i a s o m w ic is sma e oug owa s + i co e co e e asym o ic o e ies o e o igi a u c io Theorem 4 (Co e se ma i g; see ig 4). Let f (x) be continuous in 1 + co[ with Mellin transform f * (s) having a nonempty fundamental strip a i (i Assume that f* (s) admits a meromorphic continuation to the strip y for some y a with a finite number of poles there, and is analytic on 9l(s) = y Assume also that there exists a real number j e (cc, 13) such that f * (s) = 0(Isr) with r > 1 ( 3 when Is' -4 co in y s 1(s q. If f * (s) admits the singular expansion for s E y a 1 E C1 k ( ( k e A — Ph. Flajolet et al. 1 Theoretical Computer Science 144 ( 1995 ) 3-58 f *(s) f (x) o e a e m i asym o ic e a sio x - e o u s i e a sio a ig o u s i e a sio a + oo Mu i e o e oga i mic ac o 1 ( e ( og k a (s )k+1 k! 1 _ ig x ( og k a co (s ok +1 k! o e wi imagi a y a = o + it uc ua io s = e1 og egu a y s ace o es ou ie se ies i og ig 4. e u ame a co es o e ce as ec s o e co e se ma i g ( eo em a Co o a y 1 then an asymptotic expansion of f (x) at is 1 k — f (x) = E (k— )! g ( — ( k e A (ii Similarly assume that f * (s) admits a meromorphic continuation to r f e oo makes use o e i e sio eo em a o a esi ue com u a io usi g a ge ec a gu a co ou s i e e e e s i o f *(s), see ig 5 As e o e i su ices o co si e case (i co es o i g o co i ua io o e e e 9' e e se o o es i < fi>. Co si e e i eg a 1 J(T) = . f *(s)x - s s 1 7 w e e C e(T) e o es e ec a gu a co ou e i e y e segme s [11 — iT, + [ + iT, y + [y + iT, y — [y — iT, — Assume a T is a ge a 13 (so 1 o a o es so E Y y Cauc y s eo em J (T) is equa o e sum o esi ues w ic is y a i ec com u a io ( _ - 1 ( - s E k Res E d ( og k ( k e A s = ( k e A 4'k ( (k 1)! Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995) 3-58 1 7 + i + i 7 — iT — iT ig 5 e co ou C use i e oo o eo em o s e ese o es o f*(s). e ow T e s o + co e i eg a a o g e wo o i o a segme s is (T r ) a us e s o as T cc. e i eg a a o g e e ica i e 91(s = a ies i si e e u ame a s i e s o e i e se Me i i eg a w ic co e ges gi e e ec ease assum io o f* a equa s f ( y e i e sio eo em (si ce f (x) is co i uous e i eg a a o g e e ica i e 9i (s) = y is ou e y a qua i y o e o m 1 y+i s I I s = (1 (1 + y = O( 7c i _ i „ gi e e g ow assum io o f * . us i e imi J( cc equa s f (x) us a emai e e m a is ( Y us e sum o esi ues a is o e s a e o m i a og I ma y cases f* (s) is me omo ic i a com e e e o ig a - a e a sa is ies e co i io s o eo em ; e a com e e asym o ic e a sio o f * (s) esu s Suc a e a sio may e ei e co e ge o i e ge I i e ge e e a sio is y ecessi y o y asym o ic I co e ge i may i some cases e ese e u c io s e ac y u is ca o e a ge e a e ome o eci a e om e se ies a o e as f ( a f (x) + (x) a e e same asym o ic e a sio s a a co w e e e m( is a " a " u c io ike 3( = e ( + A oo o a e ac e ese a io equi es a e emai e i eg a s a o g e ica i es e o e o; is ee s a as e oug a u i o m ec ease o f*(s) a o g e ica i es 22 Ph. Flab/el et al. Theoretical Computer Science 144 ( 1995 ) 3-58 Example 5. The transform of (1 + x)'. Take v > 0, and consider the function (s (v — s) ck(s) = (v) that is analytic in the strip <0, v>. The singular expansion to the left of 91(s) = 0 is E ( — 1)k F (v + k — 1) 1 k=0 k! (v) (s + k) • The problem here is the behavior of the original function 1vi2+ico f (x) = )(s)x- ds 27E fv/2—ico as —■ e ec ease o (s along vertical lines results from the decrease of the Gamma function. Thus, the conditions of Theorem 5 are satisfied. In this way, one finds, for any integer M, ( — 1)1` (v + k — 1) k m+1/2 f (x) = E + ( k=0 ( The series on the right-hand side is a truncation of the Taylor series of (1 + x)', by virtue of the binomial theorem. We have thus proved that f (x) = (1 + + vi (x) for a function w(x) that decreases at 0 faster than any power of x. Sharper estimates of inverse Mellin integrals show that the remainder integral tends uniformly to 0 when 0 < x < 1, so that the representation is in fact exact. We have thus found indirectly the Mellin pair 1(s (v — s) f (x) f * (s) = (1 + x)" (v) Example 6. A classical divergent series. The function 0(s) = 1(1 s) . sin its is analytic in the strip <0, 1>. In 91(s) < 1, it admits the singular expansion E (— n! 1 n=0 s + n which encodes for the original function the asymptotic expansion CO f (x) E (—o n! x". n=0 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 23 In fact 4)(s) is the Mellin transform of the "harmonic integral" " e f (x) = dt 1 + whose consideration goes back to Euler. Periodicities. A pole of at a point = o- + it with a nonzero imaginary part induces a term of the form x - = x -a e -i logx which contains an oscillating component that is periodic in log x with period 27c/t. Mellin transforms are precisely useful for quantifying many nonelementary fluctu- ation phenomena. In this context, there sometimes occurs a vertical line of regularly spaced poles (see Section 6). A direct extension of Theorem 4 allows for the presence of infinitely many poles in a finite strip. Corollary 1. The conclusions of Theorem 4 remain valid assuming only a weaker form of the growth condition (23) along a denumerable set of horizontal segments 13(s)I = T ; where T— co. Proof. Use the proof of Theorem 4 but restrict T to belong to the discrete set T; which, by the growth condition imposed, must avoid the poles of f * (s). In particular regularly spaced poles of f * (s) along a vertical line correspond to a Fourier series in log x. For instance, simple poles of f* (s) at the xk = o- + 2ikn/log will arise from a component of the form (1 — /3 - s) -1 in f * (s). The corresponding residues rk induce for f (x) the infinite sum + E rk x - k = + x rk exp ( — 2iloc log B x), k Ez kEz where the sign is + for poles left of the fundamental strip (expansion at 0) and — otherwise (expansion at cc). 3. Harmonic sums This section builds upon the functional properties of the Mellin transform and the fundamental correspondence to develop a framework adapted to the analysis of harmonic sums. Definition 3 (Harmonic sums). A sum of the form G(x) = E .1k g (itk x) (25) 24 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 is called a harmonic sum. The ilk are the amplitudes, the ilk are the frequencies, and g(x) is called the base function. The Dirichlet series of the harmonic sum is the sum A (s) = E Aki k- s (26) In this paper, the frequencies either decrease to zero or increase to co. By possibly considering G(1/x) and changing the base function accordingly, we may always reduce ourselves to the single case il k + co, which we assume in this section, unless otherwise stated. The Mellin transform of a harmonic sum factors, as already mentioned in the introduction, into a product of the Dirichlet series A (s) and of the transform g* (s) of the base function. This is true unconditionally for finite harmonic sums Elementary arguments provide a first extension to infinite sums. Lemma 2. Assume that g(x) is bounded over any interval [a, b] c (0, + co), and that it satisfies g(x) = 0 (xv) when x 0 and g(x) = 0(x") when x + cc, with u > v. Assume that the Dirichlet series A (s) has a half-plane of absolute convergence 92(s) > a. that has a nonempty intersection z1 with the strip < —u, —v>. Then the harmonic sum G(x) of (25) is defined for all x in (0, + co). The transform G* (s) is well-defined in z1 where it factors as G* (s) -= A (s) • g* (s). We recall that the theory of generalized Dirichlet series [41] guarantees that a Dirichlet series like A (s) has a half-plane of absolute convergence 91(s) > aa and a half-plane of simple convergence 91(s) > er, where Proof. First consider the special case where g(x) is 0(1) at 0, is 0(x -1 ) at co, and A (s) convergence absolutely for 91(s) > — 6 for some 6 > 0. Then over (0, +oo), we have both I g (x)I < C/x and lg (x) I I G( 1 < • E I Ak I 9 by the assumption that A(0) converges absolutely. Consequently, G (x) = 0 (1) every- where and in particular at 0. By summation, G (x) is also 0(x -1 ) everywhere, and in particular as x + cc. By the dominated convergence theorem, G* (s) therefore exists in the strip <0, 1>. The general case reduces to the special case by normalizing g(x) and considering 4'(x) = x-ut(v-u)g(x 1 - o Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995 ) 3-58 25 Our treatment throughout this paper relies on the assumption that g* (s) and A (s) in Lemma 2 are continuable as meromorphic functions in regions of the complex plane larger than what their definitions imply, and additionally satisfy controlled growth conditions towards + i cc. Definition 4. (i) A function 0(s) is said to be of fast decrease in the closed strip a ...C. 91(s) a if for any r > 0, Cs = (1s1 as I s I —> co in the strip. (ii) A function is said to be of slow increase in the closed strip a -C. 91(s) a if for some r > 0, 0(s) = 0(1 s r ), as I .51 —> co in the strip. The property of fast decrease means that 0(s) decays faster than any negative power of Is I. For instance, the function 7t/sin TES is of fast decrease in any finite closed strip since the complex exponential representation yields it — 0(e s = o- + it, si i s which exhibits an exponential decrease along vertical lines. A similar exponential decay holds for F (s) by the complex version of Stirling's formula [1, p. 257]: y-1/ e — itito 0_ + co na + 27c t y (27) The property of slow increase holds for many Dirichlet series and, for a < 1, one has [83, p. 94] + it) = (1t1 1 ). (28) (Uniform versions of (27), (28) also exist, see the cited references.) The property of slow increase is also called "finite order". It means that 4)(s) is of at most polynomial growth in the strip. The function ((s) is bounded in any closed strip of the form [1 + E, M] and thus is of slow increase there. Classical theorems [83] that extend (28) near 91(s) = 1 show that it is of slow increase in any finite strip of the complex plane. The theorem below represents the basic paradigm for the analysis of harmonic sums. It presupposes that the Mellin transform of the base function is of fast decrease and the Dirichlet series of the harmonic sum is of moderate growth in an extended region of the complex plane. In the statement, only the abscissa of simple convergence is needed, a handy improvement for many applications. 26 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 h r ( a mo ic sums Consider the general harmonic sum G(x). Let the trans- form of the base function g* (s) have a fundamental strip - the half-plane of convergence of A (s) intersects the fundamental strip of g*(s): a < 13 and let a' = ma (a cc ); - the functions g* (s) and A (s) admit a meromorphic continuation in a strip y fl> and are analytic on 91(s) = y, for some y a; - on the closed strip y (s (a + / the function g* (s) is of fast decrease and the function A(s) is of slow increase. Then the harmonic sum G(x) converges for all x > on ( + co An asymptotic expansion of G(x) as x till an error term 0 ( is obtained by termwise translation of the singular expansion of G* (s) = A (s)g* (s) according to the rule A 1_ A ( —1)k - ( og k (s _ k + 1 k! r f i s se ec a a i a y i eg a io a scissa a in (a', 13) a ake some ao suc a a ao a e i e sio eo em o i es 1 +icc - E Ang(tinx)=—,, E - g (s sc s ( 9 n=1 zin —leo =1 e e is a we -k ow ou o e g ow o a (ge e a i e i ic e se ies i a y i e a - a e s ic y i e io o i s sim e co e ge ce omai a ea s C(Is + 1 o some co s a C, see [ 1 3- ] om is we a e E - g (s C(Isj + 1)• Ig*(s)l- (3 =1 wi e mi s o a y e omi a e co e ge ce eo em a es a is es e co e ge ce o G(x) e esse as a i e se Me i i eg a 1 +ico G ( = A (s)g* (s) ds (31 _ c„ I a i io e ou (3 s ows a G(x) is ( ° w e e a was c ose a i a i y i (a us e s i co N A G*(s) = lim E 2„ • g ( x) xs = im E -) • g* (s), N-0.0 5 =1 - co i i „ a is o say e u ame a e a io G*(s) = A(s)g*(s). Ph. Flajolet et al.( Theoretical Computer Science 144 (1995) 3-58 27 The proof is now completed by a straight application of the Co e se Ma i g eo em (Theorem 4). El A symmetric result holds near x co, assuming now the meromorphic extensions to hold to the left of the fundamental strip; the sign is reversed due to the orientation of the contour. This discussion is summarized by the following informal statement. Mellin summation formula. U e e co i io s o eo em 5, E Ao(iik ± E Res (g (s A(s - s se - o a e a sio ea 0, e sum is o e e se o o es o e e o e u ame a s i a e sig is +; - o a e a sio ea co e sum is o e o es o e ig o e u ame a s i a e sig is - In addition, the relation (31) shows that G(x) is of C class since A(s g (s is of fast decrease so that derivation under the integral sign is permitted. Example 7. a mo ic um e s The function [ 1 1 1 I k ( k1= k k + x i = k=1 k 1 + /k satisfies ( = where „ is the harmonic number. (It is closely related to (d/dx) log ( Its Mellin transform is It (s = C(1 - s sin i s with fundamental strip < -1, 0> and singular expansion to the right of this funda- mental strip 1 y - E s S k=1 S — k • Thus (with a minus sign corresponding to the expansion at cc) 1 —11k 131 1 1 1 1 „ — log n + y + — + L - log + y + + 2n k k 2n 12n2 1 + Generalized harmonic numbers can be analyzed in the same way, see [37, p. 300] for a table giving Ph. Flab/el et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 Example 8. Stirling's formula. om e o uc ecom osi io o e Gamma u c io o e as (x):= og ( + 1 — y = [ -c — og (1 +1] (se - +GO =1 is e am e is simi a i s i i o e e ious o e e Me i a s o m is It e (s = C( s si It wi u ame a s i — —1 e e a e ou e o es a s = —1, s = a sim e o es a e osi i e i ege s 1 — y] [ 1 og 7 ( e * (s) 1 + + + E (s + 1 + 1 s S =1 n(s + n) e ce S i i g s o mu a 2 og ( — og ( e - 1 7c + E n ( — 1 " Example 9. Polylogarithms. e u c io " e - " L(x)= E n =, is e a e o o y oga i ms a a e e i e o i ege w y i( = z'1 w E I o e as L(x)= i (e - We assume i s a w is a osi i e i ege w = k. e u c io k ( is a y ica a mo ic sum wi a s o m (s) = C(s + k)r(s), a i i 91(s > 1 a eas e si gu a e a sio o ai s om e si gu a e a - sio s o C(s) (s i e a r(s): ((k — 1 E ( = n! s + n k-1 ( _1 k— - 1 k_i (S e —oo + co (k-1)!L(s+k-1) + s+k— 1 e ou e o e a s = k — 1 i uces a oga i mic ac o a _ - E (k — n) ( [ og + k-i] + (k — 1)! = n! k-1 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 29 e ig - a si e is a io i o y a asym o ic e ese a io a i as u e e e e ese a io is e ac as see om u i o m ou s e i i g om g ow o mu ae o e Gamma a e a u c io s (( 7 ( We a e o ai e i is way e Co e - agie e ese a io o mu a o o y oga i ms [5 3 7] W e w is o a i ege e e is o co ue ce o si gu a i ies a acco i g y e oga i mic e m isa ea s o i s a ce C - 11 ( E_-- E + E ( 1 „ =1 n=0 n! Example 10. Modified theta functions. e sum e - ( = E n= nw is e a e o e a u c io s e ge e a case ca e ea e ike e o e u we es ic a e io o w = 1 e case w = 1 a ises i e a a ysis o u e so see [5 1 9 a E 5 -5] O e as ONs)=-1 F(-s )C(s+ 1 e e is a ou e o e a s = a o es o r(s/2) a e e e e ega i e i ege s is gi es e e a sio as 1 y 1 1 1 1 1 + e (s - -[- s + 2s J + 1 s + + s + y 1 4 e i (x) 1°gx ± + 1 + + • e si ua io o eo ( is i e e si ce e o es o r(s/2) a e e e i ege s a e ca ce e y e a ues o e e a u c io C ( — = C ( — = • • • = us j e ( = 0 4 — o a y M > e ege e a e asym o ic se ies ca o y e asym o ic is as e am e is e a e o e mo u a a s o ma io o e a u c io s (see a so oisso s summa io o mu a i Sec io a a ious e a e o ms a e ee i es iga e om e asym o ic s a oi y ama u a e e ui K u a Go e see [ ; 15 3; 3 ; 5 1 9] ama u a iscusses some c i e ia a e ai e e is e ce o i i e e a sio s [ C ] o simi a u c io s I ge e a e a se ce o si gu a i ies i a e a - a e i ica es a e mi a i g e a sio wi e o e ia y sma e o e m (I ma y ac ica si ua io s es ima es o e e o e m esu om a yi g e sa e oi me o o emai e i eg a s 30 Ph. Flajolet et al. 1 Theoretical Computer Science 144 ( 1995 ) 3-58 The two examples that follow are motivated by the two types of harmonic sums mentioned in the introduction. One involves strongly fluctuating amplitudes resulting though in a smooth global behavior. The other has geometrically increasing frequen- cies so that the Dirichlet series has a line of poles resulting, by Corollary 1, in an oscillating asymptotic behavior expressed by a Fourier series. Example 11. A i iso sum The analysis of the height of trees (see the introduction) suggests considering the sum ( = E (k e - kx k=1 with (k being the number of divisors of k Given the equality (k 2 =c (s k=1 i the Mellin transform of ( is (s = (s C2 (s with fundamental strip <1, + cc >. There are singularities at s = 1 s = 0, then at all the odd negative integers, so that 1 7 1 (C( — k — 1 2 1 D* (s)-- [(s — 1 2 ± s — 11 + s s= k - (2k + 1)! s + 2k + 1 . This translates into the asymptotic expansion of ( as x .— 0: 1 (C( —2k — 1))2 k+1 ( — - ( — log x + y + — — 4 x k=0 (2k + 1)! Clearly sums of divisors can be treated in a similar way as their Dirichlet series are expressible in terms of the Riemann zeta function. Ramanujan discovered a number of related formulae later treated by Berndt and Evans using Mellin transforms [8]. From the proof of Theorem 5, what essentially matters is the a a ce between the growth of A (s and the decrease of g (s The asymptotic expansion remains valid as long as G (s is globally 0(s') for some r > 1 in the extended strip (the C character mentioned above may get lost, though). Also, in accordance with Corollary 1, it is sufficient that the decrease of G (s hold along a discrete set of horizontal segments. This permits to cope with situations involving infinitely many imaginary poles, corresponding to an oscillatory behavior. Example 12. A ou y e o e ia sum a e io ici ies The prototype of harmonic sums with a fluctuating behavior is the function G( = E e- k k= Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995) 3-58 31 w ose e a io is soug as e Me i a s o m is (s G (s = (s • E( ( 1 — s = _ k 1 — s e e a e i i i e y ma y o es o A (s a e oi s ik7C k = og • e Gamma u c io ec eases as as 3(s + co e i ic e se ies A(s = (1 — - s _ s ays ou e a o g o i o a segme s a ass i e wee o es a 13(s 1 = ( k + 1 i / og o k a i ege us Co o a y 1 o eo em a ies e e e esi ue o A (s is 1/( og y i ue o e io ici y us 1 j 1 (3 — kI 7 og s — k C) G o a y e o es o G (s a e us a ou e o e a si g e o es i uce y A (s a e k a o es a e ega i e i ege s a s a i g o G ( yie s Y 1 1 ( — G ( = — og ( + + Q ( og + E og n=i — agai a e ac e ese a io e e Q ( og co e ses e co i u io om e o - e o imagi a y o es a e ec s (3 1 1 Q ( og = E r(x0x - x , =,,, E rock)e -2iknlog,x ioe -- k €1 { } - - k E { } us a e e oga i mic e m G ( co ai s a uc ua i g e m o o e (1 a is e esse as a ou ie se ies i og wi e ici coe icie s o a Gamma y e A e i e y simi a me o e mi s o e ac uc ua io s o sum ike K ( = E (1 — e k k= w e + oc see o osi io e ow We o se e a I ( + i / og 1 5 5 1 1 - a ue o e as ec ease o (s a o g imagi a y i es e uc ua i g u c io Q ( og s ays ou e y 1 - Suc mi u e uc ua io s co s i u e a commo ea u e o ma y Me i a a yses A c ose y e a e e am e is a a y e y a y ( is u c io ( i [ 37] w o o es a ama u a ai e o o ai a e eme a y oo o e ime um e eo em ecause is a gume eg ec e ese i y "wo es" a isi g om imagi a y o es 32 Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995) 3-58 4. Summatory formulae What we call here an asymptotic summatory formula is a formula that gives the asymptotic form of a class of harmonic sums when either the amplitude-frequency pair stays fixed (and the base function is taken in a wide class of functions) or the base function stays fixed (and the amplitude-frequency pair varies). The generality afforded by Theorem 5 permits to deduce a large number of summatory formulae, a typical case being generalizations of the Euler-Maclaurin formula. Euler-Maclaurin summation. The classical Euler-Maclaurin formula provides a full asymptotic expansion expressing the convergence of a Riemann sum to the corres- ponding integral. The expansion assumes the base function to be C and then involves the Bernoulli numbers. The formula has been extended by Barnes [6] and later revisited by Gonnet [36] in the case of certain functions singular at the origin. Proposition 1 (Generalized Euler-Maclaurin summation). Let g(x) be such that g(x) E ck k for some increasing sequence {ock } with occ, > - 1. Assume that g* (s) exists in - o 13> for some 13 > 1 and is meromorphically continuable to a function of fast decrease in any finite strip of < - co, 13>. Then G(x) E--- E g(kx) - g (x) dx + E ckC( - cek /k k=1 .- 0 x k =0 Proof. The transform G* (s) is g* (s) c(s) with residue at s = 1 that equals g* (1), itself the integral of g (x). By Theorem 3, there are also poles of g* (s) and G* (s) at each - ock . The result then follows by Theorem 5 after taking into account the slow increase of (s) in finite strips. Example 13. A sum with half-integer exponents. The sum 1 s = k=E 1 10 1 (n + k) 31 is an Euler-Maclaurin sum since 1 1 1 S„ = — G (-- ) where G(x) E (kx), (x) x 11 (1 + 31 k= 1 The expansion of g(x) near x = 0 involves half-integer exponents: CO k + 1\ g(x) E ck k - 11 ck = ( -1)k (k 02-2k ( k= k )* Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995) 3-58 33 I a i io g (s as u ame a s i 1/ 5/ a a e ici a s o m ( e i i g om e Eu e ia e a i eg a o om a ea ie e am e (s - 1/ 1( - s g (s = (s -1 - s = (3/ w ic gua a ees i s as ec ease us a sim e com u a io yie s 1 S„ - + E ck c( -k +1 k+ 3/ k= e a e e am es ike n 1 E k=1 k( - k mo i a e y e a a ysis o i e o a io seque ia sea c a e iscusse i [3 ] a a u a e i [37 9 ] ( e a a ysis is owe e mo e e ica e wi g (s o y ec easi g ike s -11 Example 14. A e o e ia sum wi squa e oo s e u c io G( = E e •/;;x n= a mi s e e a sio 2 1 ce _ G( - - + E c(-- k .-o x k=i k si ce i s Me i a s o m is (s C(s/ Ma y simi a summa o y o mu ae ca e gi e u e e assum io s o o osi- io 1 ike E —ok - i g (kx)— E cko — 1+ " C( - c k k k=1 k= 1 1 E ( og k g (k - og - g ( + g ( ( og k=1 CO + E ckc( —cox". k= o i s a ce ama u a gi es [ ] k og E e - IC 1 ( og - y + og / 7( + ( 3 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 Several examples of Ramanujan have been treated systematically by Berndt and Evans using Mellin transforms [9]. Functions with logarithmic singularities at 0 can also be dealt with in this way; see the papers by Barnes [6] and Gonnet [36] for alternative approaches. o e o o ma Me i a a yses In this range of problems, it is of interest to observe that asymptotic series follow o ma y from an exchange of summations in the harmonic sum, with the Dirichlet series coming out as a purely divergent series (!). This observation often constitutes a useful heuristic in the discovery of possible summation formulae then rigorously established by Mellin transforms. An illustration is provided by the formula G( = E — g(( k + 1 — E ckri( —ock ik k=1 k= where " (1 k (s = E k 1 (2k + 1)s is an entire "L-function" associated with a character modulo 4, see [12]. In effect, one has formally G(x) E — E c, k (2m + 1) k ) = E ck cck — ok(2m + k m=0 E Co ( - ak k k= The Mellin transform justifies such formal manipulations while "explaining" the occurrence of additional terms like the integral in the Euler—Maclaurin summation. Dyadic sums. Sums involving powers of 2, of the type Gw ( = E 2 g k o are particularly frequent in the analysis of algorithms and we call them ya ic sums. In applications x usually represents a large parameter. Proposition 2 (Dyadic sums). e g( e suc a g( E dk x - k - c° k= Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995) 3-58 35 g for some increasing sequence {fik } with 13 > 0. Assume that (s) exists in 12ikrc _ ik og dk -Ok GO ( E g kEE 9* k 0 log 2 ( log 2 ) e k= 1 — k Proof. The transform of G o (x) is g* (s) (s) = 1 — 2s and it is readily subjected to Theorem 5 and Corollary 1. Proposition 2 is characteristic of a large number of similar summation formulae. Confluence of singularities will in general induce logarithmic factors. Example 15. The standard dyadic sum. Consider the analysis of Go (x) when now g(co) 0 0 and g (x) E dk x - k. -"- c° k= (Thus do = g(co) 0 0.) There is now a double pole of Gt (s) at s 0 so that a two-term expansion is required for g* (s) there. By the fundamental splitting, one has ( ) f:1 g* (s) = + g (x) dx + (g (x) — g(co))xs - 1 dx, so that g (co goo dx (goo goo dx g* (s) = + [g] + 0(s) where y [g] = o The constant y[g] is called the Euler constant of g since y[e - x — 1] = — y. Thus, k G(x) g(oo) log x + g () + + P(log x) + E log 2 k=1 1 — for an explicitly determined periodic function (• This example illustrates in passing the general technique for determining terms of the series expansion of a Mellin transform outside of its convergence strip by adapting the technique of subtracted singularities of Theorem 3. Poisson's summation formula. There is a fruitful connection between the classical Poisson summation formula [43, Vol. 2, p. 271] + 1 " (2n + E f (kx) = — E f — ) , I(u) = f (t)e i u du, (33) k= 3 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 e u c io a equa io o e iema e a u c io a o e o mu ae o a a ysis ike e mo u a a s o ma io o e a u c io s ( is was i s o se e y iema i is o igi a memoi o e a ia ac io e a sio o e co a ge a co ec io ca e ase o e Me i a s o m o we -c ose a mo ic sums We o y o i e e e ie i ica io s a e e o e ai s o [55] a wo ks ci e e ei e y e o ic co a ge a mi s wo equi a e e essio s o e as a ( a mo ic sum o e o e ia s e o e as a ( a mo ic sum a ia ac io ecom osi io 1+ e 2" co (ic = 1 _ e - 2 x — 1 + E e 1 1 = _ I (3 x =, + n • e u c io a equa io o e iema e a u c io ES C(S = s cs -1 si — (1 — s) (1 — s). (35 2 a ea s o e e "image" u e e Me i a s o m o e i e i y e wee e wo o ms o (3 see [ 3 ] e oo o oisso s summa io o mu a e esu s om aki g e Me i a s o m o o si es o (33 a i se i g e u c io a equa io o C(s is a oac as e me i o es a is i g a ge e a equi a e ce e wee summa io o mu ae o e oisso y e a u c io a equa io s o i ic e se ies [7 55] a ies ook co ai s a i ec i us a io [13] wi a oo o e mo u a a s- o ma io o e e eki u c io Perron's formula. A co ec io o summa o y o mu ae usua y e e e o as e o s o mu a e ess a ia sums o coe icie s o a i ic e se ies as com e i eg a s o e i e se Me i y e a ie o e i ic e se ies i se [ 1] I ou e s ec i e e o s o mu a esu s om aki g e s e u c io H (x) o (13 as e ase u c io e a s o m o H (x) ei g H* (s) = 1/s o e as o ma y o a i ic e se ies A (s) = 2k tLk s 1 e+ico s k E k H (uk x) = A (s) x — . s t x< i e o s o mu a is esse ia i a a y ic um e eo y o i s a ce i e oo o e ime um e eo em I as ee ece y em oye o e a a ysis o cos u c io s o " i i e-a -co que " a go i ms e e i e mi s o ca u e e io ic uc ua io s a a e o e o a ac a a u e o a ica io s o me geso a ma ima i i g i com u a io a geome y see [ 5]; o igi a sums a occu i e a a ysis o se e a a go i ms (so i g e wo ks egis e a oca io we e e o [ ] A e a e c ass o i i e-a -co que ecu e ces (wi ma imum as ee Ph. Flajolet et al. 1 Theoretical Computer Science 144 ( 1995 ) 3-58 37 treated via Tauberian methods by Fredman and Knuth [35], and elementarily by Pippenger [69]. Note that the use of Perron's formula is often delicate since the transform (s does not have good decay properties towards + i co. PART II. COMBINATORIAL APPLICATIONS In this part, we survey some of the major applications of Mellin asymptotics to combinatorial problems. Sums related to Catalan numbers form the subject of Section 5, and Section 6 develops the important "Bernoulli splitting process' that leads to dyadic sums. Additional combinatorial applications form the subject of Section 7. 5. Catalan sums Catalan sums have the form n ( o (36) S = E 2n / where the Ak are of an arithmetical character. Such binomial sums occur in average values of characteristic parameters of combinatorial objects enumerated by the Catalan numbers, ( 2n (37) n + 1 n like plane trees, binary trees, or ballot sequences [11, 38]. The historic paper of De Bruijn, Knuth, and Rice [16] provided the first analysis of this type. It concerns the expected height „ of rooted plane trees of n nodes under the uniform distribution. The sequence Ak is then the divisor function (k We briefly explain here the connection between this combinatorial problem and a Catalan sum like (36). (See also [51, p. 135] for details.) A plane tree decomposes recursively as a root node to which is attached a sequence of trees. Let A n be the number of trees with n nodes; the ordinary generating function of the sequence {A„} is defined by CO A ( = E A„ " = I Then, by the classical laws of combinatorial analysis (see, e.g., [33, 38, 78, 87]), the decomposition translates into a functional equation that admits an explicit solution 1 — 1 — A ( = and A ( = 1 — A ( 2 We have A = C„ with C„ the Catalan number of (37). 3 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995) 3-58 Similarly let Ah(Z) be the generating function of trees of height at most h. As height is inherited from subtrees, one then has the basic recurrence Ah + 1(z) — with A 1 ( = 1 — A ( Thus, the A ( are rational fractions that are also approximants to an infinite continued fraction representing A ( Solving the implied recurrence yields the closed form (1 + — 4z)" — (1 — — 4z) A ( = (1 + — +1 — (1 — — 4z) +1 • It results that the Ah can be expressed in terms of A(z) alone; their Taylor expansion then derives by the Lagrange—Biirmann inversion theorem for analytic functions: An + 1,h — An+1,h— 1 = E (n (38) where ( = A ( 2n ( 2n ) 2n Om( 1 1 — ( — m + 1 — m n — m) n( —2n 1 — m) • Thus, the number of trees of height h appears as a "sampled" sum of the 2nth line of Pascal's triangle (upon taking second-order differences). By a well-known form of the expectations of discrete random variables, the mean height H + satisfies 1 i + 1 = E E i ( Ah+1 h j Grouping terms according to the value of jh then reduces this expression to a simple sum: 1 F = A E d(k)wk(n). (39) 'in+ k We are therefore led to considering sums of a pattern similar to (36), s,(°) = E ( I=1 n) since 1-.1 _ H +1 = S;(, 1) — 2Sr + s" , (40) + 1 Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995) 3-58 39 e ea me o e ce a sum ei g y ica S i i g s o mu a yie s e Gaussia a o ima io o i omia um e s o k = w a wi k = o(n 31 ), o e as ( .2nk e-w2 (1 w4 3w2 5W8 - 5w - + ( 2: 3 is ea s o i o uci g e co i uous a mo ic sum G ( = d(k)e -k X a a e eme a y a gume ( omi a io o e ce a e ms us i ies e use o e Gaussia a o ima io i si e Sr : 1 = G (—,)+ o(1 ( 1 / e asym o ic a a ysis o G( o sma ( ow = -1 / is e simi a o E am e 11 a o e om G*(s) _ 1 s (s 2 o e ge s 1 ( - og ( + 37 + - + • • • ( e o e sums S ( 1 a e ea e simi a y om ( ( 1 ( (a ei a a ogues i is ou a the expected height of a random plane rooted tree of n nodes is /ttn - + o (1 u asym o ic e a sio s cou a so e e e mi e y is ec ique e asic me o e e co sis s i a o ima i g Ca a a sums (3 y Gaussia sums o e o m CO G(x)= l — 2x2 a ea i g e a e y Me i a s o ms e a e Ca a a sums su ace i e a a ysis o a c e s o -e e me ge so i g e wo k [75] a i egis e a oca io [3 5 ] w e e e a i me ic u c io l i o e i (3 is ei e a u c io o e G ay co e e ese a io o k o e u c io (k) e ese i g e e o e o i e ime um e ecom osi io o k. 40 Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995) 3-58 6. The Bernoulli splitting process and dyadic sums e e ou i s i i g ocess is a ge e a mo e o e a om a oca io o esou ce ei e i e ime omai (suc as s a io s s a i g a commo commu ica- io c a e o i e s ace omai (suc as keys s a i g some ima y o seco a y s o age w ose a a ysis usua y ea s o a a ie y o ya ic sums e ocess akes a se G o i i i ua s a s i s em ecu si e y as o ows - I ca (G) 1 e e ocess s o s a o s i i g occu s - O e wise ca (G a eac g E G i s a coi e G o a G 1 e e wo su se s o G co es o i g o e g ou s o i i i ua s a i g i e ea s ( a ai s (1 e e ocess is ecu si e y a ie o e wo su se s G o a G 1 A ea i a io o e ocess may e esc i e y a ee (G) w ose i e a i a y o es co es o o s i i gs o mo e a 1 e eme ; e e e a o es ei e co ai a si g e i i i ua o e em y se I o e iews e e eme s o G as a i g e e e mi e a i i i e seque ce o i s e ee (G) is o i g u e digital trie associa e o G iewe as a se o "keys"; see [37 5 7 ] e ie a o a e eme g i (G is ac ie e y o owi g a access a ic a e y g. A seque ia e ecu io o e s i i g ocess a so co s i u es a way o egu a e access o a commo s a e c a e (g ou s co sis i g o si g e i i i ua s may e i e ei message wi ou i e e e ce ; is is e tree communica- tion protocol o Ca e a akis- sy ako ; see [ 3] Gi e a e ca i a i y o e o igi a g ou G is n, e e a e wo asic a om a ia es e um e I, o o i ia se a a io s ages co es o i g o e um e o i e a o es i (G); e o a um e „ o coi i i gs co es o i g o a i e a a e g i (G). e e ec a io s i„ = E {I„} a e = E {L„} sa is y ecu e ces a e ec e a u e o e s i i g ocess; o n o e as in = 1 + E nn,x(ik + in- k), en=n+ E nn,k((k + e -k k= k = k = ( k ( 3 wi i i ia co i io s io = i 1 = eo = ei = e s i i g o a i i ies tt„, k a e s eci- ic o e e ou i s i i g ocess a ey e ese e o a i i y o u i g k ea s ou o coi i s e asic ec ique o so e ( 3 co sis s i i o uci g e e o e ia ge e a i g u c io s . zn zn 1(z) = E L(z) = E en ( n = 0 • • wi w ic ( 3 a s o ms i o z 1(z) = e" 1 (;) + (e — 1 — z), L(z) = e 1 L (- ) + z(er — 1 ( 5 Ph. Flajolet et al.' Theoretical Computer Science 144 ( 1995 ) 3-58 41 A u c io a equa io o e o m ( = e / + a(z), ( wi a(z) a k ow u c io a ( e u k ow is so e y i e a io 4)(z) = a(z)+ 2ezl = a(z)+ 2ezI a (1 ) + 4e 3z 1 4) (4- ) = x = E 2kez(i - 2 - k)a(—z ). ( 7 k = 0 k is i ci e a ies o /(z) a L(z) wi a(z)= ez — 1 — z a a(z)= z(ez — 1 es ec i e y U o e a i g e e o e ia s o e i s e e ici o ms — n = E 2k [i y1 i n — y (1 — k = 1 n — In = E [1— (1— y 1 ( k = om e e e mos i ec ou e is e e o e ia a o ima io (1 — a) = e n l ( — = e - n + (n 2 e -na I is egi ima e o use i i ( (see [5 131] o a us i ica io ase o s i i g e sum Wi - / k F(x)= E 2k[1- +—x)e G(x) = x E [1— e - / k ] k = k k o e i s e eme a i y i„ = F (n) + ( i a e = G(n) + 0(j/). e u c io s F(x) a G( a e ya ic sums o a y e a ea y co si e e e ce F(x)= + ( og + ( /7 Y 1 G(x)= og + ( + + xQ ( og + ( og w e e P(u) a Q(u) a e a so u e y co e ge ou ie se ies e u i g o ies a eg ec i g e e io ic uc ua io s a a e o am i u e ess a 1 we i a e um e o i a y o es is o a e age a ou 1 a % waste in storage, w i e e a e age e en/n o a a om e e a o e is a ou og n w ic i e i o ma io - eo e ic se se co es o s o an asymptotically optimal search cost. 42 Ph. Flajolet et al.( Theoretical Computer Science 144 ( 1995) 3-58 e a ia ces o ese a ame e s a e ee e e si e y s u ie y Ki sc e o e o i ge a S a kowski [53 5 ] ese au o s o e a i is o a i ia o em o o ai e e ac o e o e omi a e ms as e e a e o - i ia ca ce a io s equi a e o mo u a y e i e i ies isco e e y ama u a acque a eg ie [ 5] a e es a is e a a e g a o e si e a e o asym o ica y o ma y is i u e e a a ysis i o es e s u y o o i ea i a ia e i e e ce equa io s o w ic e o owi g is y ica T(u, z) = uT 2 (u — ) + (1 — u (1 + e ' 2 I o ma io o a ge com e wi u ake as a a ame e ca e ga e e y a quasi- i ea i a io ocess ase o co si e i g og T(u, z) w ic is ame a e o Me i a s o m ec ique O ce is is o e e coe icie [e]T(u, z) ca e eco e e asym o ica y a si ce i is i ec y e a e o a c a ac e is ic u c io o e um e o o es e Gaussia esu is e es a is e Ma mou s ook [ ] co ai s a e ai e esc i io o is i e es i g a ica io o Me i a s o ms o o i ea i a ia e o ems A ece e e sio as ee ma e o su i ees a e em e — i a a com essio sc eme [ 7] 7. Other combinatorial examples Com i a o ia sums may ecessi a e a ce ai amou o " e ocessi g" e o e ei g e uci i e o a mo ic sums as a ea y e em i ie y Ca a a sums o sums e a e o ies is sec io esc i es i i ec a ica io s o e asym o ic a a ysis o a mo ic sums o wo ew y es o com i a o ia o ems Reduction to standard harmonic sums. o ges u s i a om i a y s i gs a e ea e y K u [57] i a a e a ea s wi e equi a e o em o ca y o aga io i a a e i a y a e s e e e o em equi es a a a ysis o omi a o es o a ami y o a io a u c io s e e ua y ea i g o ya ic sums Examples 16. Longest runs in strings. Co si e s i gs o e a i a y a a e ai = { 1} e o em is o es ima e e e ec e e g L„ o e o ges u o s i a a om s i g o e g w e e a e 2" ossi e s i gs a e ake equa y ike y e is i u io was s u ie y e e [ 1] a K u [57] e o a i i y a a a om s i g o e g as o u o k co secu i e s is 1 1 — qn = Ezn] ( 9 1 — 2z + zk +1 • e se o suc s i gs is esc i e y e egu a e essio 1 k • ( 1 k w e e 1 e o es a seque ce o ess a k s a ( e o es a i a y e e i io o a a e ; e ge e a i ci es o com i a o ia a a ysis e mi o w i e e o i a y Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 43 ge e a i g u c io o e se o s i gs u e co si e a io as 1_ k 1 1 — z 1 — z(1 — zk)/(1 — z)' w ic us i ies ( 9 e k e e sma es osi i e oo o e e omi a o o ( 9 a ies e wee a 1 A a ica io o e i ci e o e a gume s ows suc a oo o e is wi a o e oo s a a e o a a ge mo u us y omi a o e a a ysis e q , k sa is y 1- q ,k Ck(2pk) n wi Ck = (5 k ( — (k + 1)p)' o a ge n u i e k. e e omi a o o e ac io i ( 9 e a es ea z = 1/ ike a " e u a io " o 1 — so a o e e ec s k o e a o ima e y as k —0 co. A e eme a y a gume e ai e i [57] s ows a i ac Pk = 1( 1 ± 2 —k— 1 + (k - k (51 Acco i g y ck = 1 + (k y mea s o co ou i eg a io K u us i ies e use o (51 i si e (5 o a wi e a ge o a ues o k a w ic esu s i e a o ima e o mu a q k _ - k — e t k e o e e a o ima io e " to g„, k • o owi g [57] o e i s CC CO (1 ) L„ E [1- g , k ] = E -4„,d+o k k= ( 1 = E [1- e--.2-k-I+ o k = 1-1 is is a y ica case o a ya ic sum s u ie e ea e y i e ious sec io s a 7 )1 1 ik7c _ og L„= og + + E e + 1 og og k e { } ( og us e e ec e e g o e o ges u L„ uc ua es a ou og + 33 7 wi a mi u e am i u e A e i e y simi a a a ysis o i es e e ec e si e o e a ges summa i a a om com osi io o a i ege n. Nonstandard Dirichlet series. e a go i m o Probabilistic Counting i o uce i [ 7] e mi s o es ima e wi i a ew e ce e um e o is i c e eme s o a a ge i e usi g o y a e y sma amou o au i ia y memo y a is o i e es i e co e o que y o imi a io i a a ases e a a ysis a ea s o e o s a a 44 Ph. Flajolet et al.1 Theoretical Computer Science 144 ( 1995) 3-58 i ic e se ies co _ ov(i) N(s) E =1 w e e v(j) is e sum o e igi s i e i a y e ese a io o j. Example 17. Probabilistic counting. e e y esig o e a go i m o [ 7] ecessi- a es e a a ysis o a a ame e o " w e e gi = { 1} is e se o a i i i e y o g i a y s i gs e u = (u 1 ,..., u) e a e eme o ."; e a ame e = „ is e i e o ake e a ue k i e seque ces 1 1 1 1 a occu as i i ia segme s o some o e if s u o u; s a s wi e a e 1 A ausi e easo i g sugges s a e e ec a io k„ o o e " ake wi u i o m measu e s ou e c ose o og 2 u ig e -o e asym o ic i o ma io is equi e e e i s a i c usio —e c usio a gume o [ 7] s ows a 2k n {R„ k} E (-1 (1 (i _ (5 -= k I is o e i [ 7] a i a ce a egio ea og 2 o e ca use e e o e ia a o ima io (1 — a " e "a w i e simu a eous y e e i g e a ge o a ues o j o i i i y is us i ies e a o ima io co co (-0vo)e-in,k= — ( / " qn,k k w e e 4n , k = E (1- e (53 = = e i e e u c io ( ( = E ( _ 1 (i e -i = _ e - 1 (5 Eq (53 mea s a e cumu a i e is i u io g„,,, is we a o ima e y 0(n/2"). I ac a e eme a y a gume o [ 7] es a is es a °° „ E g„, k = (n) + o(1 w e e 0(x) (55) = E (y k1= k=1 a e asym o ic o m o 0(x) o a ge is equi e e u c io s 0(x) a 0(x) a e o a mo ic sums om (5 a (55 o e i s s 0* (s) s 0* (s) = N(s)• r(s), 0* (s) — = N (s) F (s). (5 1 _ s 1 — 5 I a icu a a mo e e ai e i es iga io o N (s) is ee e G ou i g e ms y i e e i i io o N(s) a usi g e i omia eo em (see a so o osi io e ow s ows a N(s) is a e i e u c io a is o mo e a e g ow i a y ig a - a e owi g o ca ce a io s a o e y e sig a e a io o e ies o e seque ce ( — 1 " ( is e mi s o us i y (55 a e ce y a esi ue Ph. Flajolet et al. 1 Theoretical Computer Science 144 ( 1995) 3-58 45 com u a io i gi es e asym o ic o m o ( y N' (0) (x) = og + + + ( og + o(1 (57 og og o some osci a i g u c io P(u) o ce mo e ou o e o sma am i u e e g ou i g ec ique yie s ( = - 1 (use i (57 a i e mi s e essi g ( as a a e a i g sum o oga i ms us om (55 (57 o e ge s R„ og (9 + ( og n) + o(1), w e e c° ( m + 1 (-1 2 - 11 ev 77351 m m e e esu s a a go i m ase o e i ea a (1/q " ca e u e i o a u iase s a is ica es ima o o e a io i u k ow um e o is i c i a y s i gs co si e e e seque ce e(j) = ( 1 is e c assica ue-Mo se seque ce i ic e se ies e a e o N(s) a e ee co si e e i [ ] Simi a ec iques a e use i o e o a i is ic es ima io a go i ms ike Approximate Counting [ ] a e co isio eso u io me o s o [39] PART III. GENERAL MELLIN ASYMPTOTICS e as sec io s a e s ow a Me i asym o ics is a ica e o a mo ic sums o i e e i ic e se ies is o mo e a e g ow w i e e a s o m o e ase u c io is o as ec ease e a ica io s e cou e e so a a e ma e a esse ia use o o e ies s eci ic o e iema e a u c io o e Gamma u c io o o e si e u c io is a s ows a a ks o ge e a eo ems ese e i Sec io Me i asym o ics may a so e a ie o "im ici " a mo ic sums w e e i is o o ge equi e o a e c ose o ms o ei e e i ic e se ies o e a s o m o e ase u c io 8. General conditions for Mellin asymptotics I is sec io we o i e ge e a co i io s u e w ic Me i a s o ms a e sma owa s + i co a esc i e a ge e a c ass o i ic e se ies a a e o mo e a e g ow ese esu s e e e a ge o a ica i i y o Me i asym - o ics o a muc a ge c ass o a mo ic sums 46 Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995 ) 3-58 Smallness of Mellin transforms. The smallness of a Mellin transform is directly related to the degree of "smoothness" (differentiability, analyticity) of the original function. Proposition 3 (Smallness in the fundamental strip). (i) Let f(x) be locally integrable with fundamental strip (ii) If in addition f(x) is of class T. and the fundamental strip of e f contains f * (a + it) = o(Itl - r) as t + co. Proof (sketch). (i) From the form f *(r + it) = f (x) e - trx e i og dx, the function f* (s) is an integrable function hashed by a complex exponential. By the Riemann-Lebesgue lemma [43, 81], f * (s) tends to 0 as t + oo (ii) The second form results from the formula i [e f (x); s] = ( - 1)1. sr f * (s). D The next proposition shows that smallness extends beyond the fundamental strip for smooth functions with smooth derivatives. Proposition 4 (Smallness beyond the fundamental strip). Let f(x) be of class W' with fundamental strip f (x) = E C k x 4 (log x)k + 0 (x 7) , (58) ( k eA where the satisfy -a < y (resp. y < - 13). Assume also that each derivative (dj/dxj) f(x) for j = 1, ...,r satisfies an asymptotic expansion obtained by termwise differentiation of (58). Then the continuation of f*(s) satisfies f * (a + it) = (I — as It] -> co (59) uniformly for a in any closed subinterval of ( -y, 13) (resp. of (a, - y)). Proof. It suffices to consider extension to the left of f * (s). Choose some positive number p > y and define a(x)=( E C k (log x)k exp ( - xP). ( k eA a o e e a I eo e ica Com u e Scie ce 1 ( 1995 3-5 47 e u c io g(x) = f(x) — a(x) sa is ies e assum io s o o osi io 3 a i s a s o m g* (s) is us o (I s I - i i s u ame a s i —y, ft>. e a s o m a* (s) is i se e o e ia y sma gi e g ow o e ies o e Gamma u c io a i s e i a i es us f * (s) = a* (s) + g* (s) sa is ies e s a e ou s A a y ici y is e s o ges ossi e o m o smoo ess o a u c io f(x); i a case e a s o m (s) ecays e o e ia y i a qua i ia e way r p t n (E o e ia sma ess i e a a y ic case Let f (x) be analytic in So where So is the sector So = {z e CIO < ItI < + (x) and I a g( I } with 0 < i Assume that f(x) = 0(x -') as — in So, and f (x) 0(x - 13) as oo in So . Then f*(c + it) = 0(e -°1 `1 ) uniformly for c in every closed subinterval of (a fl). r f (sketch). e i eg a e i i g Me i a s o ms i is case a ie o a a a y ic u c io y Cauc y s eo em e i eg a io co ou may e ake as e a - i e o s o e 0: e co f* (s) =I f We - dt o e c a ge o a ia e t = pe' gi es f* (s = eies (peio ps-i p. e esu o ows as e i eg a co e ges Sma ess e e s ou si e o e u ame a s i y a a gume simi a o a o o osi io a ase o su ac i g sui a e com i a io s o e o e ia s Ex pl 8 A sum with an implicit transform. e a mo ic sum d(n) G (x) --= E =1 / cos nx wi d (n) e i iso u c io a mi s e a s o m C (s) • g* (s) w e e g* (s) is e a s o m o g(x) = (cos 1/ e u c io g*(s) is o sim y e essi e i e ms o s a a u c io s (i is "im ici y" e i e as e a s o m o g(x)). owe e e u ame a co es o e ce o i es i s com e e si gu a e a sio (see E am e w i e o osi io s a 5 s ow a g* (s) is o as ec ease i a y ig a - a e us Me i a a ysis a ies a e asym o ics o G (x) as o ai s Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995) 3-58 om e si gu a e a sio s o g (s) a 4 (s). O e ge s log x (g*)'(1) + 2yg*(1) 1 G (x) - g* (1) + + + ( m -o o a y M e e g* (1) = g (x)dx a (g*)' (1) is a "Eu e co s a " o g (x) w ic is e essi e as a i eg a as iscusse i E am e 15 Example 19. A lattice sum. e goa e e is o iscuss ce ai ( - ime sio a " a ice sums" o e o m F (x) = E E ow ( m =1 =1 (See a so [1 ] o e a e e e o me s Suc a sum may e ew i e as Ek r (x) f (kx ), w e e r(k) is e um e o e ese a io s o k as a sum o wo squa es e co es o i g i ic e se ies is e " r (k) 1 P(s) = E is = (m2 ± n s k = 1 is a e e y i g es s o i s e a io a + i co a i s s ecia a ues e 0 (x) = e-M2X2. e Me i a s o m o ( is 1 s ( = p(s) • - ew - A a ea y iscusse Me i a a ysis o e( yie s i s asym o ic e a sio ea = 7C 1 ( 1 = + R(x), ( 1 i — e emai e e m R(x) ei g e o e ia y sma ( is a e a i e y esu s om oisso s o mu a us ( a mi s a e ici ee- e m asym o ic e a sio t i 1 e ( 11 = — - — + - + ( 2x 2x 4 wi agai ( e o e ia y sma Co seque y p(s) is me omo ic i e w o e o C wi o es a s = 1 o y a si gu a e a sio 7C 1 + 1 (s s=o + [0] s - 2 + [0], = 2(s - 2) - s _ + • • 3 I o e o a ow o e e sio s o ig e ime sio s o ig e owe s we o o make use e e o s ecia o e ies o p (s). See [ C I ] o a i i accou o ese as ec s Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995) 3-58 9 I a i io o osi io 5 qua i ies e g ow o e a s o m o e ( i e o m e (n14 — °I t us (s is i se o g ow a mos e s o a y e i a y i i e s i o e com e a e is k ow e ge e mi s o a a y e a ice sums o e o m ( o i s a ce 1 ( = E (m + + p 3 sa is ies ( = (1/ o i e o e akes ( = (1 + -3 O e as i 3 (s — 1 (s — (s = si i s e e esu s a asym o ic o m o ( a i uces a co es o i g es ima e o ( I 37c 5 — IC ( — 1 5 4p e e o e m ei g e o e ia y sma si ce (s a is es a e e e ega i e i ege s e me o c ea y e e s o ig e ime sio a sums Growth of Dirichlet series. i ic e se ies w ose com o e s Ak uk a mi esce - i g asym o ic e a sio s i owe s o k a e e o e y o ei g me omo ica y co i ua e wi we i i i ua e o es Proposition 6 (G ow o s ecia i ic e se ies e Ak a i k a mi asym o ic e a sio s i esce i g owe s o k as k co " a A ( E — k w (1 + E — r 0 O = e e i ic e se ies Ek Ak k- S ca e co i ue o a me omo ic u c io A(s i e w o e o e com e a e e u c io A(s is o mo e a e g ow i a y ig a - a e o e com e a e Proof (ske c e a icu a case a = = gi es e esse ia i ea e i omia eo em yie s b1A 1s ( s(s +± 1)/bl )2 1 ...... 1 — = — [ao + + • • • [1 „ 1 = a E k(s ws k nws +k ( o a ami y o o y omia s k y summa io o e i s a A(s as i a y i i e s i e same si gu a i ies as a a ia sum o (ws + E k(S (WS k k1 50 Ph. Flajolet et al. Theoretical Computer Science 144 (1995) 3-58 (This is easily justified rigorously by terminating forms of the binomial theorem.) Also, A (s) is of moderate growth since each component zeta function shares this characteristic. In the general case, one finds similarly (with a slight abuse of notations) A (s) C(ws + oco + E P(s)Nws + , (63) e where ranges over the set S of all linear combinations of the form r S i + ••• + For instance, from the expansion 1 ( 2n ri i + 1 4" n ) 1/ 8n 128n one finds by summation that 112n 1 1 [c (s ± 1) 1 + 3) + 1 A (s) = C(s + + (s 1 = i 'VA n )ns 2) 8 2 128 2 (64) where R(s) converges in the same half-plane as C(s + -I), namely in ( + oo). In a similar vein, if Q(u) is a polynomial that does not vanish at the positive integers, and P (u) is an arbitrary polynomial, then P ( A (s) = E = (Q (Os has poles only at a discrete set of rational numbers that is bounded from the right. Amplitudes involving the harmonic numbers lin = 1 ± 1/2 + • • • + 1/n can be treated in this way by considering fi = Hn — log n. From 1 1 1 = log n + y + + + 0 (n - ), 1 1 one deduces A (s) = '(s) + yC(s)+ (s + 1) — (s + 2) + C(s + 4) =i n (s e ( — 5, co>) 1 y 1 1 (s — 1) s — 1 i s = ± 2s 12(s + 1) ± 120(s + 3) For instance, A (s) has a simple pole with residue 1/120 at s = — 3. More information on this function may be found in [5]. Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995 ) 3-58 51 Example 20. A sum with binomial amplitudes. e u c io ( 1 1 ( 1 . 3 1 ( 1 3 5 1 G (x) = 2 ) 1 + 1 + 1 + 3 as e a s o m A (s) i /si it s w e e A (s) = E„ _ ( - s e si gu a i ies o A(s) a e gi e y Eq ( e ce 1 G(x) : - + A(0) + - i + ( w e e A(0) is o ai e y e a a y ic co i ua io ec ique o o osi io " 1 1 1 _ (1 A(0) = E1 ; [( E _ + A u asym o ic e a sio i owe s o )0 12 ca e e e mi e i is way Example 21. Stirling's formula for modified Gamma functions. Co si e e o em o e a ua i g asym o ica y as + oo H(x) = (1 + n=i n(n + 1 )) O e es ima es i s ea e a mo ic sum G(x) = og H (x) a is a a mo ic sum wi a s o m a i i - 1 -1 ; 1 EG* (s) = A( -s) • ( .7C ) w e e A (s) = s si ITS n1 (n + n)s • om o osi io A(s) as a si gu a e a sio i uce y e e a sio o (1 + n -1 ) - s. O e as A(s)= ( s - ( s + 1 + R(s) w e e R(s) co e ges ike ( s + i 91(s - e Me i a s o m G* (s) us as a sim e o e a s = -1- a a i e o e a is o i es "S i i g s o mu a" o H (x): H (x) -C, -ec w e e "S i i g s co s a " C is e e C = 1/11 I is a icu a case H (x) is e ici y e essi e i e ms o igo ome ic u c io s ( y e i i i e o uc o mu a o cos ( ; e e am e is o y mea o emo s a e e ge e a a oac o e a a ysis o suc mo i ie Gamma u c io s 52 Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995 ) 3-58 A similar technique works for Dirichlet series A (s) whose elements Ilk, Pk admit expansions in powers of 2k• Assume that 1+ E a 2 - ki Pk — k ( 1 E 1) '). i= 1 i= 1 A technique entirely similar to that of Proposition 6 applies. From 2 -ks,( 1 (a1 _ i s)2 -k k + (a +(b — 2a 1 b 1 — 2b )s + ibls2)2- one gets by summation 2 - s 2-5-1 A (s) = 1 — 2 - s + (a1 is' 1 — 2 - s -1 ± •••• Thus, A (s) admits a pole at all points of a half-lattice 2ikic (m) k Xk = — m + for m 0, ey, og and is in addition of controlled growth away from poles in any finite strip of C. An immediate case of application is to modified dyadic sums involving frequencies of the form 1( + 1. This technique also applies to sums related to the Bernoulli splitting model of Section 6; for instance, the Mellin transform of oo x F (x) = [1— (1 k= 1 is F * (s) = A (s) • F (s) where A (s) = E (log (1 — 2 ') - 1 ) - s. k=1 By the technique above one finds 25 s 2s -1 A (S E — 2>). 1 — 2s 2 1 — 2s -1 This permits to compute complete asymptotic expansions without a recourse to the exponential approximation. The half-lattice of poles implies the presence of fluctuat- ing functions at each level of the asymptotic expansion of F (x). The transfer technique. Our final example illustrates the general approach seen so far in conjunction with an important transfer technique that consists in going back and forth between various harmonic sums involving related amplitude and frequencies but different base functions. This "zigzag" method appeals to the common occurrence of a Dirichlet series in two harmonic sums E ak f (kx) and Eakg (kx) , Ph. Flajolet et al. I Theoretical Computer Science 144 ( 1995 ) 3-58 53 wi g (x) = e " ayi g a s ecia o e ecause o e e a io i es a is es wi o i a y ge e a i g u c io s e a s e ec ique as a ea y ee use im ici y o i s a ce e s u y o e a mo ic sum e (1 — 1 yie s o e ies o (s) (E am e a ca i u e use o a a y e e e a u c io E k e -k (E am e 1 a e e ua y a ice sums (E am e 19 e i ci e is as o ows Assume a e e a io o F (x) = E ak f (kx) k = 1 is soug is equi es k ow e ge o e i ic e se ies ak a (s) = E k = 1 I e o i a y ge e a i g u c io A (x) = E ak x k , k= 1 is k ow a i a i io A (x) is a a y ic a = 1 ( is o e a e s o a e a i g se ies wi e eme a y coe icie s e A(e - = E b „,xm , o w ic co es o s o a si gu a e a sio b,„ di (A (e s) = a (s) (s) E (s E C m=om+s Com a iso wi e si gu a e a sio o e o uc a (s) F (s), ame y a( —m 1 a(s ‹ ( 1 m m = m! s-Fm' s ows y i e i ica io a a( —m = ( —1 mm bm = ( — 1)'" m! [e] A (e ( 5 I is way coe icie s i ma y e a sio s ca e e e mi e e ici y I ge e a e o mu ae o ai e a e o i ia si ce e o igi a se ies e i i g a (s) s o s ei g co e ge a ega i e i ege s e a s e ocess o e o i es a igo ous cou e a o o ma com u a io s wi u e y i e ge se ies See e ema ks e a i e o Eu e —Mac au i summa io s i Sec io a e eo y o " egu a i e " sums a o uc s i [ 9] 54 Ph. Flajolet et al. Theoretical Computer Science 144 ( 1995 ) 3-58 Example 22. An alternating sum with harmonic numbers. e i e G ( = E - Hk e -k x , k=1 a a ises as a a o ima io o e com i a o ia sum ( 2n E _ k k= 1 n - k ) e Me i a s o m o G( is G* (s) = h(s)- -21 F (-2S ) w e e h (s) = E - - k k=1 us G (s) as o es a = - - a e e mi i g e esi ues o G (s) equi es e values o h(s) a e ega i e e e i ege s e i ic e se ies h(s) is a e i e u c io is esu s om a sim e e e sio o o osi io o a e a i g se ies as a e a i g e a u c io s a e e i e e s ecia a ues o h(s) a e e o ai e om e e a sio E -1 k I k e -k = og k=1 1 +1- e' + e — log 2 (I log 2 + + — 2 4 4 16 1 og 1 + - — + — 3 — + 32 48 8 so a h(0) = og h( - = h( -4) = - 1 ••• us i a y G( og - — ••• • 9. Conclusions is a e as emo s a e e asic ec o ogy o Me i asym o ics o a - mo ic sums e e a c ucia o e is aye y e se a a io o e y a e u ame a co es o e ce e me o is ike y o e a ica e as soo as e e essio s i o e a e o a a a y ic c a ac e (i e o se se o e e m a o i g a a y ic co i ua io w ic is e mos im o a equi eme o e me o Com i a o ia e essio s a e us e y a u a ca i a es o Me i asym o ics e a e ec iques i co ec io wi i i e i e e ces a e e o e i [3 ] Ph. Flajolet et al. / Theoretical Computer Science 144 ( 1995) 3-58 55 As me io e i Sec io e o s o mu a e o gs o e ga a y o Me i - e a e ec iques We a e o e e o e is c assica as ec e e as i is e e si e y iscusse e sew e e I [ 5] i is s ow ow i i e-a -co que ecu e ces ea o uc ua io s o a ac a a u e a ca e qua i ie ia a Me i a a ysis; igi a seque ces a e s u ie a o g simi a i es i [ ] A i e es i g o s i g o Me i asym o ic o a mo ic sums is e c ass o t - t me o s I ma y cases a ge e a i g u c io ca e a a y e asym - o ica y as a a mo ic sum i e com e ea m ea a si gu a i y o a c i ica oi I e ecomes ossi e o es ima e e coe icie s o e ge e a i g u c io y ei e si gu a i y a a ysis o e sa e- oi me o o a co u c io o Me i a sa e- oi me o s e s a a e am e is e ui s io ee i g wo k o e e ume a io o i a y a i io s [1 ] w ic was o ows y e ge e a a oac o Mei a us i e eo y o i ege a i io s [3] Si gu a i y a a ysis use i co u c io wi Me i asym o ics may e use as a asis o e a a ysis o Ca a a sums [ ] as we as ma y o e com i a o ia sums We o e o e u o ese ques io s i a u u e a e Acknowledgements is wo k was a y su o e y e ES I asic esea c Ac io o e E C o 71 1 (A com II e au o s wou ike o a k ai e Kem (w o i o uce o e o us o e su ec i 197 o e Se gewick ( o a o g- as i g i e ac io eo Gui as a A ew O y ko ( o ume ous iscussio s o K u a G e ui ( o ei su o e mu o i ge oge e wi is g ou o co a o a o s a e ec ica U i e si y o ie a Mi ei e eg ie i i e acque a Wo ek S a kowski a co i u e a ua e ee ack a a ious s ages We a e ee y g a e u o em o ei coo e a io References [1] M A amowi a I A S egu Handbook of Mathematical Functions ( o e ew Yo k 1973 ; a e i o e 1 a io a u eau o S a a s e i io 19 [ ] A ouc e a Co e i ic e se ies a cu ious i i i e o uc s Bull. 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