DOCSLIB.ORG

Transformation of Some Lambert Series and Cotangent Sums

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Transformation of Some Lambert Series and Cotangent Sums mathematics Article Transformation of Some Lambert Series and Cotangent Sums Namhoon Kim Department of Mathematics Education, Hongik University, 94 Wausan-ro, Mapo-gu, Seoul 04066, Korea; [email protected] Received: 29 July 2019; Accepted: 8 September 2019; Published: 11 September 2019 Abstract: By considering a contour integral of a cotangent sum, we give a simple derivation of a ¥ 2pint transformation formula of the series A(t, s) = ∑n=1 ss−1(n)e for complex s under the action of the modular group on t in the upper half plane. Some special cases directly give expressions of generalized Dedekind sums as cotangent sums. Keywords: Lambert series; cotangent sum; modular transformation; Dedekind sum MSC: 11F99; 11F20 1. Introduction s For s 2 C, let ss(n) = ∑djn d be the sum of positive divisors function. For t in the upper half plane H, consider ¥ ¥ e2pint A( s) = (n)e2pint = ns−1 t, ∑ ss−1 ∑ 2pint . (1) n=1 n=1 1 − e A(t, s) is an entire function of s for every t 2 H and a Lambert series in q = e2pit for every s 2 C. The study of transformation of A(t, s) under the action of the modular group at + b t 7! a b 2 SL(2, Z) (2) ct + d c d has been a classical subject. Since A(t, s) is manifestly invariant under translation t 7! t + b for b 2 Z, one only needs to consider transformations (2) with c > 0. The main result of this article is the H a b following transformation formula of A(t, s). For t 2 , s 2 C and c d 2 SL(2, Z) with c > 0, at + b 1 (ct + d)−s A , s = A(t, s) + 1 − (ct + d)−s z(1 − s) ct + d 2 ! (3) cs−1 sec ps Z c−1 p jd iz j + 2 (−z)s−1 c + p i ct + d z − p p − z s/2 ∑ cot cot d 8(ct + d) C j=0 c ct + d c where C is the Hankel contour that encloses the nonnegative real axis in the clockwise direction but not any other poles of the integrand (see Definition1 and Theorem1). Many previously known transformation formulas can be derived as special cases of (3) by considering some particular values of s (see Corollary1). When s is an even positive integer, A(t, s) appears in the Fourier series of the holomorphic Eisenstein series of weight s and satisfies a simple transformation law [1,2] (Corollary1 (i) and (ii)). The case s = 0 is closely related to the Dedekind eta-function ¥ h(t) = epit/12 ∏ 1 − e2pint (t 2 H) n=1 Mathematics 2019, 7, 840; doi:10.3390/math7090840 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 840 2 of 10 as a branch of log h on H is given by pit log h(t) = − A(t, 0). (4) 12 We can thus derive the transformation law of log h(t) by setting s = 0 in (3) (Corollary1 (iii)). Among many other proofs of the eta-transformation formula, we mention the work of Siegel [3], who gave a 1 simple proof of it under t 7! − t by considering a certain contour integral of a product of cotangent functions. This method was generalized by Rademacher to the full modular group in [4]. The transformation property of A(t, 0) also has many applications; for example, it is used in Rademacher’s derivation of an analytic formula of the partition function [5], which improved the result of Hardy and Ramanujan [6]. The formula also brings out the notion of the Dedekind sum, which has interesting arithmetic properties [7]. In Corollary1 (iii), we obtain the cotangent sum representation of the Dedekind sum directly from (3). The transformation formula of A(t, s) when s is an even negative integer, which led to the idea of generalized Dedekind sums, was found by Apostol [8]. A computational error in [8] was corrected by Mikolás [9] and Iseki [10]. We obtain this formula in Corollary1 (iv), also with cotangent sum expressions for the generalized Dedekind sums. The general transformation property of A(t, s) for complex s was first studied by Lewittes [11] in connection with certain generalized Eisenstein series. Berndt derived a transformation formula involving an integral expression in [12], which is further generalized in [13]. We note that the formula (3) involves an integral expression different from the formula in [12]. Studying the behavior of an integral of a more general cotangent sum may be an interesting topic for further investigation. Throughout this work, logarithms and powers are taken with the principal argument. 2. Transformation Formula p Definition 1. Let c, d 2 Z with (c, d) = 1 and c > 0. For t 2 H and s 2 C, we denote r = ct + d and define ! cs−1 Z c−1 jd iz j ( ) = (− )s−1 + − − Ic,d t, s s z c ∑ cot p irz cot p dz (5) 8r C j=0 c r c where C is the Hankel contour that encloses the nonnegative real axis [0, ¥) in the clockwise direction excluding any other poles of the integrand. Consider the integrand in (5). Since e2piz cot pz = −i 1 + 2 , (6) 1 − e2piz cot pz ! −i exponentially as Im z ! ¥. Since both ir and i/r are in the upper half plane, the integrand in (5) decays exponentially as Re z ! ¥ and thus (5) is an entire function in s. We also note that for s 2 Z, the integral (5) along C reduces to the integral around the origin, and Ic,d(t, s) = 0 whenever s is an odd integer as the integrand becomes even. We write the following standard argument as a lemma. Lemma 1. For Re s < 0, ! pics−1 c−1 jd iz j I (t, s) = Res (−z)s−1 cot p irz − cot p − (7) c,d 4rs ∑ ∑ c r c z2Cn[0,¥) j=0 where the sum denotes the sum of all residues in Cn[0, ¥). Mathematics 2019, 7, 840 3 of 10 Proof. Let KN = CN + SN be the keyhole contour in Cn[0, ¥), where CN is the part of C in Definition1 in the region jzj ≤ N, and SN traverses along the circle jzj = N. We can choose a sequence N ! ¥ such that each KN stays well away from the poles of the integrand of (5). The integral over CN is 2pi times the sum of the residues inside KN minus the integral along SN. The lemma follows since the integral along SN vanishes as N ! ¥ for Re s < 0. Under this − assumption, the sum of the residues in Cn[0, ¥) is absolutely convergent, and the term (−z)s 1c of the integrand can be ignored as it is holomorphic in the slit plane. We now restate and prove (3) using the notation introduced above. H a b Theorem 1. For t 2 and s 2 C, let A(t, s) be the series given by (1). For g = c d 2 SL(2, Z) with c > 0, we have 1 ps (ct + d)−s A (gt, s) = A(t, s) + 1 − (ct + d)−s z(1 − s) + sec I (t, s) (8) 2 2 c,d at+b where gt = ct+d , Ic,d(t, s) is given by (5) and z(s) is the Riemann zeta-function. Proof. Since both sides of (8) are entire in s, it suffices to prove the equality for Re s < 0. We use Lemma1 to compute Ic,d(t, s). For each 0 ≤ j < c, the factor cot p(irz − jd/c) of (7) has poles at z = (k + jd/c)/(ir) for k 2 Z with residue 1/(pir). Hence the contribution to Ic,d(t, s) from the residues of (7) at these poles, excluding z = 0, is given by s−1 s−1 ic 0 1 jd 1 jd 1 j − k + cot p k + − (9) 4rs ∑ ir c ir c r2 c k2Z, 0≤j<c where the prime in the sum indicates that the term with (k, j) = (0, 0) is omitted. Since (c, d) = 1, every integer can be uniquely written in the form kc + jd for k 2 Z and 0 ≤ j < c. Let −n = kc + jd. Since ad ≡ 1 mod c, we have na ≡ −j mod c, and (9) equals − ics−1 1 n s 1 n na − + s ∑ cot p 2 . (10) 4r n6=0 ir irc r c c We now use the fact that cot z is odd. For n positive, s−1 s−1 s−1 s−1 1 n 1 n n − pis pis −s ps n −s − − = e 2 + e 2 r = 2 cos r , ir irc ir irc c 2 c and by (6), (10) becomes 0 1 2pi − n + na 1 ps e r2c c B ( − ) + s−1 C 2s cos Bz 1 s 2 ∑ n C . (11) 2r 2 @ n>0 2pi − n + na A 1 − e r2c c Furthermore, since n na n(at + ad−1 ) at + b − + = c = n , r2c c ct + d ct + d the sum in (11) is in fact 2pin at+b e ct+d at + b ns−1 = A , s . ∑ 2pin at+b + n>0 1 − e ct+d ct d Mathematics 2019, 7, 840 4 of 10 We now treat the other poles in a similar fashion. For 0 ≤ j < c, the factor cot p(iz/r − j/c) of (7) has poles at z = (k + j/c)(r/i) for k 2 Z with residue r/(pi).
Load more

Recommended publications
  • Limiting Values and Functional and Difference Equations †
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan; [email protected] * Correspondence: [email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions.
    [Show full text]
  • Overpartitions and Functions from Multiplicative Number Theory
    U.P.B. Sci. Bull., Series A, Vol. 83, Iss. 3, 2021 ISSN 1223-7027 OVERPARTITIONS AND FUNCTIONS FROM MULTIPLICATIVE NUMBER THEORY Mircea Merca1 Let α and β be two nonnegative integers such that β < α. For an arbi- trary sequence fa g of complex numbers, we consider the generalized Lambert series n n>1 P αn−β αn−β n 1 anq =(1 − q ) in order to investigate linear combinations of the form P > S(αk − β; n)ak, where S(k; n) is the total number of non-overlined parts equal k>1 to k in all the overpartitions of n. The general nature of the numbers an allows us to provide connections between overpartitions and functions from multiplicative number theory. Keywords: partitions, Lambert series, theta series MSC2020: 11P81 05A19. 1. Introduction A partition of a positive integer n is a sequence of positive integers whose sum is n. The order of the summands is unimportant when writing the partitions of n, but for consistency, a partition of n will be written with the summands in a nonincreasing order [2]. An overpartition of a positive integer n is a partition of n in which the first occurrence of a part of each size may be overlined or not [6]. Let p(n) denote the number of overpartitions of an integer n. For example, p(3) = 8 because there are 8 overpartitions of 3: (3); (3)¯ ; (2; 1); (2¯; 1); (2; 1)¯ ; (2¯; 1)¯ ; (1; 1; 1); (1¯; 1; 1): Since the overlined parts form a partition into distinct parts and the non-overlined parts form an ordinary partition, the generating function for overpartitions is given by 1 X (−q; q)1 p(n)qn = : (q; q) n=0 1 Here and throughout this paper, we use the following customary q-series notation: ( 1; for n = 0, (a; q)n = (1 − a)(1 − aq) ··· (1 − aqn−1); for n > 0; (a; q)1 = lim (a; q)n: n!1 Because the infinite product (a; q)1 diverges when a 6= 0 and jqj > 1, whenever (a; q)1 appears in a formula, we shall assume jqj < 1.
    [Show full text]
  • Generating Special Arithmetic Functions by Lambert Series Factorizations
    Volume 14, Number 1, Pages 31{45 ISSN 1715-0868 GENERATING SPECIAL ARITHMETIC FUNCTIONS BY LAMBERT SERIES FACTORIZATIONS MIRCEA MERCA AND MAXIE D. SCHMIDT Abstract. In this collaborative article, we summarize the known use- ful and interesting results and formulas we have discovered so far. Sum- marizing results from two related articles by Merca and Schmidt, we arrive at so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and open problems related to the inverse matrix entries de- fined in Schmidt's article and the Euler partition function p(n) which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article. 1. Introduction 1.1. Lambert series factorization theorems. We consider recurrence relations and matrix equations related to Lambert series expansions of the form [4, x27.7] [1, x17.10] n X anq X (1.1) = b qm; jqj < 1; 1 − qn m n≥1 m≥1 + + for prescribed arithmetic functions a : Z ! C and b : Z ! C where bm = P djm ad. There are many well-known Lambert series for special arithmetic functions of the form in (1.1). Examples include the following series where µ(n) denotes the M¨obiusfunction, φ(n) denotes Euler's phi function, σα(n) denotes the generalized sum of divisors function, λ(n) denotes Liouville's function, Λ(n) denotes von Mangoldt's function, !(n) defines the number of distinct primes dividing n, and Jt(n) is Jordan's totient function for a fixed Received by the editors October 9, 2017, and in revised form January 3, 2018.
    [Show full text]
  • Rapidly Converging Formulae for $\Zeta (4K\Pm 1) $
    RAPIDLY CONVERGING FORMULAE FOR ζ(4k 1) ± SHUBHO BANERJEE AND BLAKE WILKERSON Abstract. We provide rapidly converging formulae for the Riemann zeta L s n − n function at odd integers using the Lambert series q(s)= n∞=1 n q /(1 q ), s = −(4k ± 1). Our main formula for ζ(4k − 1) convergesP at rate of about √15π 4π e− per term, and the formula for ζ(4k + 1), at the rate of e− per ≈ π3√15 term. For example, the first order approximation yields ζ(3) 100 + √15π 9 4 √15π 10 e− h 4 + √15 sinh( 2 )i which has an error only of order 10− . 1. Introduction In a recent paper [3] we analyzed the q 1− asymptotic behavior of a Lambert series of the type | | → ∞ nsqnx (1.1) Lq(s, x)= , s C, 1 qn ∈ n=1 X − with q < 1 and x > 0. At x = 1, the Lambert series Lq(s, x) is the generating | | function for the divisor function σs(n), the sum of the sth powers of divisors of an integer n [1]: ∞ n (1.2) σs(n) q = Lq(s, 1). n=1 X For odd positive integer s values, the Lambert series is related to the Eisentein series 4k ∞ n arXiv:1803.03291v1 [math.NT] 9 Mar 2018 (1.3) E2k(q)=1 σ2k 1(n) q , − B k − 2 n=1 X that are modular in nature [2]. In this paper we focus on the Lambert series at negative odd integer values of s = (4k 1), k = 1, 2, 3, ..
    [Show full text]
  • On Generalizations of Lamberts Series
    ON GENERALIZATIONS OF LAMBERTS SERIES TOM OSLER AND ABDUL HASSEN Abstract. The classical Lamberts series makes it possible to generate many remarkable transformations of series. These Lamberts series are all constructed from the function z/(1 − z). In this paper we show how to generalize these series by using an arbitrary function in place of z/(1 − z). Series transformations exhibiting beautiful symmetry are obtained. In addition, a double contour integral is found which represents these series. Our method is compared to a general procedure introduced by MacMahon. Keywords. 1. INTRODUCTION Let ∞ n B(z) = bnz , n=1 X then the series ∞ b L(z) = n (1.1) 1 − zn n=1 X is called a Lamberts series. These series are mentioned briefly in the classical texts by Abramowitz and Stegun [1], Bromwich [13], Crystal [16], Hardy and Wright [28], Knopp [36], MacMahon [39], Polya and Szego [42], and Titchmarsch [48]. With all the bn = 1, the series is an example of an analytic function defined inside the unit circle, which cannot be continued to a larger domain (see Titchmarsch [48]). Starting with the classical paper by Knopp [35] in 1913 and continuing to the present day, these series have entered into the theory of numbers, the theory of Weierstrasss elliptic functions, and the theory of basic hypergeometric series. Much of this research was motivated by the ideas of Ramanujan. Lambert Series also occur in the expansion of Eisenstein series, a particular kind of modular form. Agarwal [2] gives an excellent survey of these results. The series (1.1) can be transformed into another series ∞ ∞ b L(z) = n = B (zn) (1.2) 1 − zn n=1 n=1 X X and can be expanded in a Taylors series ∞ L(z) = b zn (1.3) d n=1 X Xd|n where the inner sum is over all divisors d of n.
    [Show full text]
  • Extended Higher Herglotz Functions I. Functional Equations 2
    EXTENDED HIGHER HERGLOTZ FUNCTIONS I. FUNCTIONAL EQUATIONS ATUL DIXIT, RAJAT GUPTA AND RAHUL KUMAR Dedicated to Professor Don Zagier on the occasion of his 70th birthday Abstract. In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F (x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F (x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study Fk,N (x), an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by Fk,N (x). Radchenko and Zagier gave a 1 log(1 + tx) beautiful relation between the integral dt and F (x) and used it to evaluate this Z0 1+ t integral at various rational as well as irrational arguments. We obtain a relation between Fk,N (x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of Fk,N (x) and some generalized Lambert series are also obtained along with other supplementary results. Contents 1. Introduction 1 2. Main results 5 2.1. Functional equations for extended higher Herglotz functions Fk,N (x) 6 2.3. Asymptotics of the extended higher Herglotz functions 8 2.4. An interesting relation between a generalized polylogarithm function and the extended higher Herglotz functions 9 2.5.
    [Show full text]
  • Exact Formulas for the Generalized Sum-Of-Divisors Functions)
    Exact Formulas for the Generalized Sum-of-Divisors Functions Maxie D. Schmidt School of Mathematics Georgia Institute of Technology Atlanta, GA 30332 USA [email protected] [email protected] Abstract We prove new exact formulas for the generalized sum-of-divisors functions, σα(x) := dα. The formulas for σ (x) when α C is fixed and x 1 involves a finite sum d x α ∈ ≥ over| all of the prime factors n x and terms involving the r-order harmonic number P ≤ sequences and the Ramanujan sums cd(x). The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when r > 1 and are related to the generalized Bernoulli numbers when r 0 is integer-valued. ≤ A key part of our new expansions of the Lambert series generating functions for the generalized divisor functions is formed by taking logarithmic derivatives of the cyclotomic polynomials, Φ (q), which completely factorize the Lambert series terms (1 qn) 1 into n − − irreducible polynomials in q. We focus on the computational aspects of these exact ex- pressions, including their interplay with experimental mathematics, and comparisons of the new formulas for σα(n) and the summatory functions n x σα(n). ≤ 2010 Mathematics Subject Classification: Primary 30B50; SecondaryP 11N64, 11B83. Keywords: Divisor function; sum-of-divisors function; Lambert series; cyclotomic polynomial. Revised: April 23, 2019 arXiv:1705.03488v5 [math.NT] 19 Apr 2019 1 Introduction 1.1 Lambert series generating functions We begin our search for interesting formulas for the generalized sum-of-divisors functions, σα(n) for α C, by expanding the partial sums of the Lambert series which generate these functions in the∈ form of [5, 17.10] [13, 27.7] § § nαqn L (q) := = σ (m)qm, q < 1.
    [Show full text]
  • Lambert Series and Ramanujan
    Proc. Indian Aead. Sei. (Math. Sci.), Vol. 103, No. 3, December 1993, pp. 269-293. Printed in India. Lambert series and Ramanujan R P AGARWAL Department of Mathematics and Astronomy, University of Lucknow, Lucknow 226007, India MS received 28 December 1992 Abstract. Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas. Keywords. Basic hypergeometric series; Lambert series; elliptic functions; mock theta functions. 1. Introduction The topic of this article called the Lambert series is a very well-known class of series, both as analytic function and number theories. Our present interest in these series emerges out of the works of Ramanujan who has profusely and elegantly used these series in a variety of contexts in his mystic research work. A number of mathematicians, later, in an attempt to unravel the mysticism underlying some of the unproved identities of Ramanujan, have also used these series with advantage. To appreciate the effectiveness of choice of the use of these series by Ramanujan and subsequently, by other mathematicians in proving a variety of identities, one must examine how beautifully they enter into the theory of numbers, theory of Weierstrass's elliptic functions and the basic hypergeometric theory. We, now present our work starting with the definition and then the impact of Lambert series on the development of the above areas.
    [Show full text]
  • Möbius Inversion from the Point of View of Arithmetical Semigroup Flows
    Biblioteca de la Revista Matematica´ Iberoamericana Proceedings of the \Segundas Jornadas de Teor´ıa de Numeros"´ (Madrid, 2007), 63{81 M¨obiusinversion from the point of view of arithmetical semigroup flows Manuel Benito, Luis M. Navas and Juan Luis Varona Abstract Most, if not all, of the formulas and techniques which in number theory fall under the rubric of “M¨obiusinversion" are instances of a single general formula involving the action or flow of an arithmetical semigroup on a suitable space and a convolution-like operator on functions. The aim in this exposition is to briefly present the general formula in its abstract context and then illustrate the above claim using an extensive series of examples which give a flavor for the subject. For simplicity and to emphasize the unifying character of this point of view, these examples are mostly for the traditional number theoretical semigroup N and the spaces R or C. 1. Introduction The “M¨obiusInversion Formula" in elementary number theory most often refers to the formula X X n (1.1) fb(n) = f(d) () f(n) = µ(d)fb ; d djn djn where f is an arithmetical function, that is, a function on N with values typically in Z, R or C; the sum ranges over the positive divisors d of a given 2000 Mathematics Subject Classification: Primary 11A25. Keywords: M¨obiusfunction, M¨obius transform, Dirichlet convolution, inversion formula, arithmetical semigroup, flow. 64 M. Benito, L. M. Navas and J. L. Varona n 2 N, and µ is of course the M¨obiusfunction, given by 8 µ(1) = 1; <> (1.2) µ(n) = 0 if n has a squared factor, > k : µ(p1p2 ··· pk) = (−1) when p1; p2; : : : ; pk are distint primes.
    [Show full text]
  • Factorization Theorems for Hadamard Products and Higher-Order
    FACTORIZATION THEOREMS FOR HADAMARD PRODUCTS AND HIGHER-ORDER DERIVATIVES OF LAMBERT SERIES GENERATING FUNCTIONS MAXIE D. SCHMIDT GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF MATHEMATICS 117 SKILES BUILDING 686 CHERRY STREET NW ATLANTA, GA 30332 USA [email protected] [email protected] Abstract. We first summarize joint work on several preliminary canonical Lambert series factorization theorems. Within this article we establish new analogs to these original factorization theorems which characterize two spe- cific primary cases of the expansions of Lambert series generating functions: factorizations for Hadamard products of Lambert series and for higher-order derivatives of Lambert series. The series coefficients corresponding to these two generating function cases are important enough to require the special due attention we give to their expansions within the article, and moreover, are significant in that they connect the characteristic expansions of Lambert se- ries over special multiplicative functions to the explicitly additive nature of the theory of partitions. Applications of our new results provide new exotic sums involving multiplicative functions, new summation-based interpretations of the coefficients of the integer-order jth derivatives of Lambert series gener- ating functions, several new series for the Riemann zeta function, and an exact identity for the number of distinct primes dividing n. 1. Introduction 1.1. Lambert series factorization theorems. In the references we have proved several variants and generalized expansions of Lambert series factorization theorems arXiv:1712.00608v1 [math.NT] 2 Dec 2017 of the form [3, 5, 6, 9] n a qn 1 n = s a · qn, 1 − qn (q; q) n,k k n 1 ∞ n 1 k=1 X≥ X≥ X and of the form n a¯ qn 1 n = s a · qn, 1 − qn (q; q) n,k k n 1 ∞ n 1 k=1 X≥ X≥ X e e Date: Tuesday 5th December, 2017.
    [Show full text]
  • The Handbook of Mathematical Functions
    Chapter 27 Functions of Number Theory T. M. Apostol1 Notation 638 Additive Number Theory 644 27.1 Special Notation ............. 638 27.13 Functions ................. 644 27.14 Unrestricted Partitions .......... 645 Multiplicative Number Theory 638 Applications 647 27.2 Functions ................. 638 27.15 Chinese Remainder Theorem ....... 647 27.3 Multiplicative Properties ......... 640 27.16 Cryptography ............... 647 27.4 Euler Products and Dirichlet Series ... 640 27.17 Other Applications ............ 647 27.5 Inversion Formulas ............ 641 Computation 648 27.6 Divisor Sums ............... 641 27.18 Methods of Computation: Primes .... 648 27.7 Lambert Series as Generating Functions . 641 27.19 Methods of Computation: Factorization . 648 27.8 Dirichlet Characters ............ 642 27.20 Methods of Computation: Other Number- 27.9 Quadratic Characters ........... 642 Theoretic Functions ............ 649 27.10 Periodic Number-Theoretic Functions .. 642 27.21 Tables ................... 649 27.22 Software .................. 649 27.11 Asymptotic Formulas: Partial Sums ... 643 27.12 Asymptotic Formulas: Primes ...... 644 References 649 1California Institute of Technology, Pasadena, California. Acknowledgments: The author thanks Basil Gordon for comments on an earlier draft, and David Bressoud for providing xx27.12, 27.18, 27.19, and 27.22. Copyright c 2009 National Institute of Standards and Technology. All rights reserved. 637 638 Functions of Number Theory π(x) that counts the number of primes not exceeding x. Notation It can be expressed as a sum over all primes p ≤ x: X 27.2.2 π(x) = 1: 27.1 Special Notation p≤x Gauss and Legendre conjectured that π(x) is asymp- (For other notation see pp. xiv and 873.) totic to x= log x as x ! 1: d; k; m; n positive integers (unless otherwise x 27.2.3 π(x) ∼ : indicated).
    [Show full text]
  • Arxiv:1602.01085V1 [Math.NT] 2 Feb 2016 12 Li (1.2) Function Rithm at Ih0 with (1.1) Series
    LAMBERT SERIES AND Q-FUNCTIONS NEAR Q=1 SHUBHO BANERJEE AND BLAKE WILKERSON L ∞ s kx − k Abstract. We study the Lambert series q(s,x) = Pk=1 k q /(1 q ), for all s ∈ C. We obtain the complete asymptotic expansion of Lq(s,x) near q = 1. Our analysis of the Lambert series yields the asymptotic forms for several related q-functions: the q-gamma and q-polygamma functions, the q- Pochhammer symbol, and, in closed form, the Jacobi theta functions. Some 1 3 ≃ 213/32π −1/π ≃ −π3/4 typical results include Γ2( 4 )Γ2( 4 ) log 2 and ϑ4(0,e ) 2πe , with relative errors of order 10−25 and 10−27 respectively. 1. Introduction We analyze the q 1− asymptotic behavior of a Lambert series and its as- sociated functions: the→ q-Pochhammer symbol, the q-gamma and q-polygamma functions, and the Jacobi theta functions. This section provides the definitions for all the functions analyzed in this paper and outlines the relationships that connect these functions to each other. Lambert series. We study Lambert series of the type ∞ ksqkx (1.1) L (s, x)= , s C, q 1 qk ∈ Xk=1 − with 0 q < 1 and x > 0. This Lambert series is closely related to the polyloga- rithm function≤ ∞ zk (1.2) Li (z)= s C, z < 1 s ks ∈ | | kX=1 through the equation arXiv:1602.01085v1 [math.NT] 2 Feb 2016 ∞ Li (qn+x)= L ( s, x). s q − n=0 X At x = 1, the Lambert series Lq(s, x) is the generating function for the divisor th function σs(n), the sum of the s powers of divisors of an integer n [1]: ∞ n (1.3) σs(n) q = Lq(s, 1).
    [Show full text]