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Moroccan J. of Pure and Appl. Anal. (MJPAA) Volume 4(2), 2018, Pages 122–157 ISSN: Online 2351-8227 - Print 2605-6364 DOI 10.1515/mjpaa-2018-0012

Harmonic numbers, harmonic and zeta

Ahmed Sebbar 1,a

Abstract. This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the hypothesis. We give an account on the of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series Mis(z) = ∞ µ(n) n ∑ s z . A significant result is that the real part f of n=1 n µ(n) e2inπθ ∑ n is an example of a non-trivial real-valued continuous function f on the real line which is 1-periodic, is not n odd and has the property ∑h=1 f (h/k) = 0 for every positive k.

2010 Mathematics Subject Classification. Primary: 11E45, 11A25, 11A41, 11R47. Key words and phrases. Harmonic numbers, Mobius¨ function, Zeta functions, Explicit Formulas, Sums of squares.

1. Introduction

This is an expanded version of two lectures delivered at Chapman University, initially devoted to the of Guy Robin and Jeffrey Lagarias concerning the formulation of the in terms of harmonic numbers and the sum of divisors, but which very quickly overflowed to various topics of , all or almost all of which are around the arithmetical Mobius¨ function µ(n) and Bernoulli numbers. This is why these notes are too fragmentary and they reflect, maybe, the taste and the views of the author. To keep the paper to a reasonable size only very few proofs are given. The paper contains nine sections (some are very short). I aimed to include many topics, but I had to omit many of them because of the limitation of time and space. The first section is mainly connected with the difference equation f (x + 1) − f (x) = g(x). Almost every property of Bernoulli

Received February 20, 2019 - Accepted April 7, 2019. c The Author(s) 2017. This article is published with open access by Sidi Mohamed Ben Abdallah University. 1Chapman University, Schmid College of Science and Technology One University Drive Orange, CA 92866 and Institut de Math´ematiquesde Bordeaux, Universit´eBordeaux I, 351 Cours de la Lib´eration33405 Talence Cedex. a e-mail: [email protected] and [email protected] .

122 HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 123 numbers or come from this equation. We will show that, conversely, Bernoulli numbers intervene well to express certain forms of solutions to the difference equation. The third and fourth sections deal with some subseries. We mention only few examples for which we give the exact sum. Some of these examples are borrowed from the theory of zeta functions of number fields. The fifth section is devoted to the so-called explicit formulas. Some properties of the are presented, since they are the generating series of the arithmetical functions that are the sums of divisors. This section also contains the formulations of the theorems of G. Robin and J. Lagarias. These two results were the first motivations behind the two lectures and finally these notes. We end this section by Vacca formula and a relation giving the Euler constant γ as an of a very lacunary series used intensely in the dynamics of paper folding. The very short sixth section deals with some probabilistic number theory. Developing more this theme would have required voluminous additions, in time and space. The seventh section is probably the richest of this work. It uses advanced properties of the , as well as the theorem. One finds in particular an interesting result on continuous functions f on (0, 1) of which all the sums of Riemann, corresponding to regular subdivisions are equal to zero:  h  ∑ f = 0, k = 1, 2, 3 . . . . 0

2. Harmonic series

eD − 1 2.1. The formal operator and the difference equation. The harmonic numbers, the related Bernoulli D numbers or Bernoulli polynomials and the are all related to the difference equation. There are several approaches to this equation and we choose the method by Fourier . For suitable f , we define Z ∞ Z 0 ixw ixw F+(w) = f (x)e dx, F−(w) = f (x)e dx 0 −∞ Proposition 2.1. The difference equation for given b < a f (x + 1) − f (x) = g(x) (2.1) has as solution 1 Z ia+∞ χ(w) + G (w) 1 Z ib+∞ χ(w) + G (w) √ + −ixw √ − −ixw f (x) = − e dw − − e dw 2π ia−∞ e iw − 1 2π ib−∞ e iw − 1 where χ(w) is holomorphic in the strip b < =w < a. The terms involving χ(w) merely represent a function of period 1, which is obviously part of the solution, a solution of the homogeneous equation. Hence 1 Z ia+∞ G (w) 1 Z ib+∞ G (w) √ + −ixw √ − −ixw f (x) = p(x) + − e dw − − e dw 2π ia−∞ e iw − 1 2π ib−∞ e iw − 1 where p(x) is any function of period 1.

All the computations are valid in L2 if for example g(x)e−c|x| ∈ L2(R) for some real c > 0. Furthermore solutions, for general g(x) without any growth condition, were given by Guichard, Appell, Hurwitz, Picard (and many others) [45]. Some solutions, related more or less to Bernoulli polynomials, exist. To be more precise we consider the difference equation (2.1) with the second member g an entire function such that the restriction to the 124 A. SEBBAR real line is in the Schwartz space S(R). We can solve this equation by right iteration, left iteration or localization. d We denote by D the differentiation operator . By Taylor’s formula the equation (2.1) takes the form dx   eD − 1 f (u) = g(u) or   1 − e−D f (u) = g(u − 1). The formal inverse of the operator (1 − e−D) is ∑ e−nD so that the equation (2.1) has the solution n≥0

f1(u) = ∑ g(u − m) m≥1 which is in fact an entire function, the series being a series of holomorphic functions on C uniformly convergent on compact sets of the .   The equation (2.1) written now in the form 1 − eD f (u) = −g(u) can be solved formally by right iteration

f2(u) = − ∑ g(u + m) m≥0 and we obtain an another entire solution of (2.1). The third method to solve (2.1) is by localization. We define the operator D−1 by Z u D−1g(u) = g(x)dx. 0 d This is a branch of the inverse of the differentiation operator D = . The equation (2.1) takes the form dx  eD − 1  Z u f (u) = g(x) dx. (2.2) D 0 z The function is holomorphic in the disk centered at the origin and of radius 2π. It has the power series ez − 1 expansion z ∞ b = n zn, |z| < 2π. z − ∑ e 1 n=0 n! 1 1 1 The coefficients bn are the Bernoulli numbers, b0 = 1, b1 = − 2 , b2 = 6 , b3 = 0, b4 = 30 ... (see (4.1) for another prsentation). A formal solution of the equation (2.1) is

−1 bn n −1 f3(u) = D g(u) + ∑ D D g(u). n≥1 n!

The difference f1(u) − f2(u) is a solution of the homogenous equation H(u + 1) − H(u) = 0 so it is a periodic function of period 1. The harmonic numbers are defined by 1 1 1 H = 1 + + + ··· + n 2 2 n or equivalently by 1 H = 1, H = H − + . 1 n n 1 n Their generating series is log(1 − x) − = H xn, |x| < 1. 1 − x ∑ n HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 125

The harmonic series is by definition 1 1 1 H = 1 + + + ··· + + ··· ∞ 2 2 n There are several proofs for the divergence of this series. We retain here the observation that for positive m, n with m < n, 1 1 1 S = + + ··· + m,n m m + 1 n is never an integer, and that Sn,2n is a , so that 1 1 1 Z 1 dx lim Sn,2n = lim + + ··· + = = log 2 = 0.69314718056 ··· n→∞ n→∞ n n + 1 2n 0 1 + x

However the (γn)n≥1, γn = Hn − log n converges to the Euler constant γ, with the following expansion 1 1 1 1 γ = γn − + + − + ··· 2n 12n2 120n4 252n6

3. Sub-series of the harmonic series

3.1. Kempner series. An interesting sub-series of the harmonic series is

0 1 ∑ n where the prime indicates that the sum is over all positive n, whose decimal expansion has no nines. The series was first studied by A. J. Kempner in 1914. The series is interesting because of the counter-intuitive result that, unlike the harmonic series, it converges. Baillie showed that, rounded to 20 decimals, the actual sum is 22.92067 66192 64150 34816. The previous result could be related to a very open of Erdos¨ on arithmetic progressions): Conjecture 1 (Erdos)¨ . Suppose that A = {a1 < a2 < ···} 1 is an infinite sequence of integers such that ∑ = ∞. Then A contains arbitrarily long arithmetic progressions. ai The origin of this conjecture is a result of van der Waerden stating that if the integers are partitioned into finitely many subsets, one of these contains arithmetic progressions of arbitrary finite length. Erdos¨ conjectured that any subset of the integers of positive asymptotic density would possess arithmetic progressions of arbitrary finite length. Roth showed that was the case for arithmetic progressions of length 3. Erdos¨ conjecture is proved by Szemeredi´ in [54]. A different proof is given by Furstenberg in [21] using ergodic theory. Szemeredi’s´ method is combinatorial and makes use of van der Waerden’s theorem. Gowers [23] gave still another proof using both Fourier analysis and combinatorics (see Remark (5.1)).

3.2. Madelung’s constants. The electric potential Vi of all ions of a lattice felt by the ion at position ri [13] zj Vi = K ∑ j6=i rij where rij = |ri − rj| is the distance between the i-th and the j-th ions. In addition, zj is number of charges of the j-th ion and K is a physical constant. If the mutual distances rij are normalized to the nearest neighbor distance r0 the potential may be written K zjr0 K Vi = ∑ = Mi r0 j6=i rij r0 126 A. SEBBAR with Mi being the Madelung constant of the i-th ion zj Mi = ∑ . j6=i rij/r0

The electrostatic energy of the ion at site ri is the product of its charge with the potential acting at its site e2 Eel,i = zieVi = . 4πe0r0 Mi It can be shown [13] that for NaCl, the Madelung constants are ∞ j+k+l ∞ j2+k2+l2 0 (−1) 0 (−1) M = −M = = Na Cl ∑ 2 2 2 1/2 ∑ 2 2 2 1/2 j,k,l (j + k + l ) j,k,l (j + k + l ) The prime indicates that the term j = k = ` = 0 is omitted. We will return to the analytic properties of this series later. An alternative formulation is ∞ n r3(n) MNa = ∑ (−1) √ (3.1) n=1 n with r3(n) the number of ways in which the integer n can be written as the sum of three squares. Legendre’s three-square theorem states that a can be represented as the sum of three squares of integers n = x2 + y2 + z2 if and only if n is not of the form n = 4a(8b + 7) for integers a and b [25]. The for r3(n) is given by ∞ n 3 2 3 4 5 6 8 9 ∑ r3(n)x = ϑ3(x) = 1 + 6x + 12x + 8x + 6x + 24x + 24x + 12x + 30x + ..., n=0 where ϑ3(x) is one of the four Jacobi theta functions ∞ n2 ϑ3(x) = 1 + 2 ∑ x . n=0 Theorem 3.1. (Emersleben [18]) The series in (3.1) ∞ r (n) ∑ (−1)n 3√ n=1 n diverges. We give a proof, different from [8], supported by the following equivalent behaviors

4 3 3 ∑ r3(n) = πx 2 + O(x 4 ), (3.2) 1≤n≤x 3 r (n) √ ∑ 3√ = 2πx + O( x). (3.3) 1≤n≤x n r (n) Indeed if the series (3.1) converges, then 3√ tend to 0. The Cesaro` sums n n 1 r3(k) cn = ∑ √ n 1 k tends also to 0. Hence 1 ( ) 3 ∑ r3 k n 2 1≤k≤n tend to 0 which contradicts (3.2). The problem of the computation of the number rk(n) of representations of n by k squares lies at the crossroads HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 127 of many ideas, analytic, algebraic and geometric (confer section (6)). The sequence r2(n) is particularly elegant. Similar to the relations (3.2) and (3.3) we have for r2(n) [27], [4]; ∞ √ r2(n) √  ∑ r2(n) = πx + x ∑ √ J1 2π nx (3.4) 1≤n≤x n=1 n r (n) √ ∞ r (n) √ ∑ √2 = 2πx x + ∑ 2√ sin 2π nx (3.5) 1≤n≤x x − n n=1 n The ordinary Bessel function Jν(z) of order ν is defined by ∞ (−1)m(z/22m+ν J (z) = , |z| < ∞. ν ∑ ( + + ) n=0 m!Γ m 1 ν The presence of Bessel’s functions is simply fascinating and mysterious.

4. Some known formulas and Hilbert polynomials

Some known (and less known) series are 1 1 1 π2 + + + ··· = 12 22 32 6 1 1 1 π4 + + + ··· = 14 24 34 96 1 1 1 π3 − + − · · · = 13 23 33 32 1 1 1 5π5 − + − · · · = . 15 25 35 1536 These values are all related to the analytic properties of zeta functions of number fields, finite degree field extensions of the field of rational numbers Q, even that the equalities ∞ 1 (2π)2n = (−1)n−1 B , (4.1) ∑ 2n ( ) 2n k=1 k 2 2n ! where B2n are the Bernoulli Numbers, given by z z ∞ z2n + − 1 = B , |z| < 2π, (4.2) z − ∑ 2n ( ) e 1 2 n=1 2n ! can be merely shown by classical methods. A very interesting arithmetical property of Bernoulli Numbers is given by Theorem 4.1 (Von Staudt-Clausen). 1 B2n + ∑ ∈ Z. p prime p p−1|2n We give an overview of the method used to prove these formulas. We choose, as an example, the third. Let N be a natural number and (Z/NZ)× the group of units in the ring Z/NZ. A Dirichlet character χ, modulo N, is by definition a multiplicative groups homomorphism χ : (Z/NZ)× −→ C×. We define the Dirichlet L-function, with respect to χ by ∞ χ(n) ( ) = L s, χ ∑ s , n=1 n 128 A. SEBBAR where χ(n) is defined as χ(n mod N) if n and N are relatively prime, and 0 otherwise. The third formula can be written 1 1 1 π3 − + − · · · = L(3, χ ) = 13 23 33 4 32 where × × χ4 : (Z/4Z) = {1 mod 4, 3 mod 4} −→ C

χ4(1 mod 4) = 1, χ4(3 mod 4) = −1. With this character (the only non-trivial Dirichlet character modulo 4) we get the classical

(−1)n L(s, χ ) = = β(s). 4 ∑ ( + )s n=0 2n 1

We define a sequence of rational functions hn(t)n≥1 by 1 + t h (t) = , (4.3) 1 2(1 − t)  d n−1 h (t) = t h (t), n ≥ 1, (4.4) n dt 1

d The operator ϑ = t , sometimes called Theta operator or Boole’s operator, is very interesting and it is possible dt d to make explicit the powers ϑn as differential operators in , with monic coefficients, that is as elements of dt d C[t][ ] [51]. Actually we have: dt Proposition 4.1. For every smooth function f of the variable t

d f h 1n−1 2n−1 i d2 f ϑn f =t + + t2 dt 1 − 2 2 − 1 dt2 h 1n−1 2n−1 3n−1 i d3 f + + + t3 (1 − 2)(1 − 3) (2 − 1)(2 − 3) (3 − 1)(3 − 2) dt3 h 1n−1 2n−1 3n−1 + + + + (1 − 2)(1 − 3)(1 − 4) (2 − 1)(2 − 3)(2 − 4) (3 − 1)(3 − 2)(3 − 4) 4n−1 i d4 f + t4 (4 − 1)(4 − 2)(4 − 3) dt4

+ ···

h 1n−1 2n−1 3n−1 + + + (1 − 2)(1 − 3) ··· (1 − n) (2 − 1)(2 − 3) ··· (2 − n) (3 − 1)(3 − 2)(3 − 4) ··· (3 − n) nn−1 i dn f + tn (n − 1)(n − 2) ··· (n − (n − 1)) dtn

This is a local problem and by the Stone-Weierstrass theorem, we restrict ourselves to monic functions f (x) = xk, k ∈ Z+ as test functions. It is equivalent to show the following lemma: HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 129

Lemma 4.1. For each real a and positive integer n: h 1n−1 2n−1 i an =a + + a(a − 1) 1 − 2 2 − 1 h 1n−1 2n−1 3n−1 i + + + a(a − 1)(a − 2) (1 − 2)(1 − 3) (2 − 1)(2 − 3) (3 − 1)(3 − 2) h 1n−1 2n−1 3n−1 + + + + (1 − 2)(1 − 3)(1 − 4) (2 − 1)(2 − 3)(2 − 4) (3 − 1)(3 − 2)(3 − 4) 4n−1 i + a(a − 1)(a − 2)(a − 3) (4 − 1)(4 − 2)(4 − 3)

+ ···

h 1n−1 2n−1 3n−1 + + + (1 − 2)(1 − 3) ··· (1 − n) (2 − 1)(2 − 3) ··· (2 − n) (3 − 1)(3 − 2)(3 − 4) ··· (3 − n) nn−1 i + a(a − 1)(a − 2) ··· (a − (n − 1)). (n − 1)(n − 2) ··· (n − (n − 1))

The Hilbert polynomials Hj(X) are extensions of binomial coefficients [47]: X X(X − 1) ··· (X − j + 1) H (X) = 1, H (X) = = . (4.5) 0 j j j!

The family (Hj(X))0≤j≤n is an algebraic basis of the space Rn[X] of polynomials of degree at most n. For each n ∈ Z+ we have n X = ∑ Anj Hj(X) (4.6) j≥0 with Anj = 0 j > n.

Several formulas for Akj are known, in particular [47] (p.2-4) n! Anj = ∑ (4.7) l1!l2! ··· lj! li>0,l1+l2+···lj=n or j   m j n j n Anj = ∑ (−1) (j − m) = ∆ X |X=0 (4.8) m=0 m where ∆ is the difference operator ∆ f (x) = f (x + 1) − f (x). This ends the proof of the lemma (4.1). It is possible to give another approach to the problem of the calculation of (xD)n f by integral representations, at least for the class of analytic functions of exponential type, in order to explain and to fix our ideas. We recall the following theorem of Wigert type [28]. Theorem 4.2. In order that f (z) should be an integral function of order 1 and type γ it is necessary and sufficient that f (z) should be of the form 1 Z f (z) = ezuφ(u) du (4.9) 2iπ C where C is a contour which includes the circle {|u| = γ} and φ is holomorphic in the open disc {|u| < γ}. In this case we have the inversion formula Z ∞ φ(u) = f (x)exu dx, γ < u. (4.10) 0 130 A. SEBBAR

Remark 4.1. An example of this situation is given by the following integral Z f (z) = ezσΦ(σ) dσ, Γ where Γ is a curve of finite length and Φ is continuous on Γ. The f (z) is an entire function of exponential type at most one and its Borel transform is Z Φ(σ) φ(u) = dσ. Γ u − σ Hence the Borel transform of f (z) is the Cauchy transform of Φ(σ). It is holomorphic outside Γ and vanishing at ∞. d The formula (4.9) gives, with D = : z dz Z n 1 n zu (zDz) f (z) = (zDz) e φ(u) du. (4.11) 2iπ C n zu This integral representation has the advantage to reduce the problem to the computation of (zDz) e which is a classical one. We introduce the exponential polynomials Φn by n x x (xDx) e = Φn(x)e . They admit the generating series: ∞ n x(ez−1) z e = ∑ Φn(x) . (4.12) n=0 n! These polynomials satisfy many identities, through derivations of (4.12) with respect to x or to z. The equality (4.11) becomes the following integral representation Z n 1 zu (zDz) f (z) = Φn(zu)e φ(u) du 2iπ C which is a kind of a multiplicative convolution. Remark 4.2. If we were interested by Cauchy type representation formula, we could use 1 ∞ kn (zD )n = zk. z − ∑ k+1 u z k=1 u A particular case is, for n ∈ Z+ z ∞ (zD )n = knzk. (4.13) z − ∑ 1 z k=1 d The inverses of the Theta operator ϑ = x are enough interesting. A possible one is given by dx Z x du ϑ−1F(x) = F(u) 0 u acting on functions F for which the integral converges. An easy induction shows that for n ∈ Z+ z (zD )−n = Φ(z, n) (4.14) z 1 − z where ∞ zk ( ) = Φ z, s ∑ s k=1 k 0 is the classical function. The formulas (7.1) and (7.2) extended by defining (zDz) as the identity operator z give (zD )n for each n ∈ Z. z 1 − z HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 131

 d p On the other hand if we could obtain for the operator z3 the equivalent of the proposition (4.1), then we could obtain dz  d p the same result for the operator by using the following relation, due to Glaisher [56], (p.109): zdz  d p 1  d p 1 zp+1 = z3 z dz zp+1 dz z2p−2 For each positive integer n we have in (4.4): 1 h (t) ∈ Qt, . r 1 − t If x ∈ C \ Z and t = e2iπx we have for (4.3) 1 1 N  1 1  i h1(t) = − lim + = cot πx, (4.15) →+ ∑ + − 2 2iπ N ∞ −N x n x n 4 and differentiating r times, r ≥ 2  1 r 1 h (t) = (r − 1)! − . (4.16) r ∑ ( + )r 2iπ n∈Z x n An application of (4.15) and (4.16) gives [35] (p.86): Theorem 4.3. Let N ≥ 1 be a natural number, χ a Dirichlet character modulo N, and r a natural number. Assume that r 2iπ/N χ(−1) = (−1) . If we put ζN = e then we have 1  −2iπ r 1 L(r, χ) = χ(a)h (ζa ). (r − 1)! N 2 ∑ r N a∈(Z/NZ)× This gives the desired equality 1 1 1 1  −2iπ 3   π3 − + − · · · = h (i) − h (i3) = . 13 23 33 (3 − 1)! 4 3 3 32

4.1. Analytic methods. The Riemann zeta function is the defined by the absolutely con- vergent for 1 1 ( ) = ζ s ∑ s . n≥1 n It is known that the zeta function extends to a on all of C, with a unique simple pole at s = 1 with residue 1. The Laurent expansion of the Riemann zeta function ζ(s) at its pole s = 1 can be written in the form 1 ∞ (−1)n ζ(s) = + γ (s − 1)n − ∑ n s 1 n=0 n! with N logn k logn+1 N γn = lim ( − ). N→∞ ∑ + k=1 k n 1 γ0 = γ is the well known Euler constant, and, for n = 1, γn is sometimes called generalized Euler constants ([5], Entry 13, p.80). Moreover, the of ζ(s) satisfies the

− s s − 1−s 1 − s π 2 Γ( )ζ(s) = π 2 Γ( )ζ(1 − s). 2 2 The function (s − 1)ζ(s) is entire and has the Hadamard’s factorization [55] (p. 30): 1 (log 2π−1− γ)s   s   s e 2 s 1 (log 2π−1)s  s  − s s (s − 1)ζ(s) = 1 − e ρ = e 1 + e 2n 1 − e ρ , ( s + ) ∏ ∏ ∏ 2Γ 2 1 ρ ρ 2 n>0 2n ρ ρ 132 A. SEBBAR where ρ describes the non trivial zeros of the function ζ. We have the Dirichlet series expansion for its inverse 1 µ(n) = , ( ) ∑ s ζ s n≥1 n where the Mobius¨ arithmetical function is defined as follows:  1 if n = 1    µ(n) = 0 if n has one or more repeated prime factors    (−1)k if n is the product of k prime factors. The , in its simplest form, asserts that x π(x) = ] {p prime, p < x} ∼ . log x For example consider x = 1021 . We know from a computation of X. Gourdon based on earlier work of M. Deleglise,´ J. Rivat, and P. Zimmermann [14] (p.11 and the references therein) that π(1021) = 21127269486018731928. The prime Number theorem is equivalent to a property on the harmonic series µ(n) ∑ = 0. (4.17) n≥1 n The Riemann zeta function plays an important role in prime number theory, arising because of the famous formula, which expresses ζ(s) as a product over primes, in the right half plane { 1}. In fact ∞ 1 ( ) = = ( − −s)−1 ζ s ∑ s ∏ 1 p n=1 n p prime There are several formulas for π(x). For instance in 1898 H. Laurent [38] gave the following formula for the counting function: ( − ) n 2iπ k 1 ! e k − 1 π(n) = 2 + , ∑ −2iπ 1 k=5 e k − 1 but this formula does not seem to be of any help in giving an asymptotic of π(x). The is an analogue of the Riemann zeta function, studied by Glaisher [22], [20]. It is defined as the following infinite series, which converges for <(s) > 1

1 1 1 1 1 1 ( ) = = + + + + + ··· P s ∑ s s s s s s p ∈ primes p 2 3 5 7 11 The series converges absolutely for for 1. It can be shown that P(s) can be continued analytically to the strip 0 <

The Euler product expansion for the Riemann zeta function ζ(s) implies that

P(ns) log ζ(s) = ∑ , n>0 n which by Mobius¨ inversion gives log ζ(ns) P(s) = ∑ µ(n) . n>0 n According to (4.1) the values P(2k) are given series involving Bernoulli numbers. For example [15]:

log ζ(2n) µ(n)  (2π)2n  P(2) = µ(n) = log |B | ∑ ∑ ( ) 2n n>0 n n>0 n 2 2n ! = 0.452247420041065498506543364832247 . . . , whence π2 ζ(2) = = 1.644934066848226436472415166646025 ··· 6 When s goes to 1, we have  1  P(s) ∼ log ζ(s) ∼ log . s − 1 More precisely [20] 1 P(1 + e) = log + C + O(e) e with ∞ µ(n) C = ∑ log ζ(n) = −0.315718452 ··· n=1 n Actually there exists a constant C ∈ R such that, for any x ≥ 3, we have the Mertens formula 1 ∑ = log log x + O(x) 1≤p≤x p or, more precisely 1 ∑ = log log x + C + O((log x)−2). 1≤p≤x p

Several properties of the function P(s) are given in [9]. It is denoted there by ζP(s). Let P the set of all prime numbers. It is an infinite set. Then π(x) := ] {p ∈ P, p < x} 1 is the number of primes less than x with added when x is prime. The n-th prime number p is asymptotically 2 n n log n. For, if pn is the n-th prime number and π is the prime-counting function we have π(pn) = n. Now, the Prime number theorem implies that p n ∼ n , log pn 1 which shows that log p ∼ log n and hence p ∼ n log n. The probability for an integer n to be prime is . n n log n 134 A. SEBBAR

5. Explicit formula

5.1. According to Riemann. The following formula, due to Riemann (1860), involving the integral Z x dt function Li(x) = relates the distribution of prime numbers to the non-trivial zeros ρ of the zeta function. 0 log t More precisely Riemann considered another weighted prime counting function, denoted by Π(x), related to the harmonic series and defined by 1 Π(x) = ∑ . pr0 n the sum being finite. By inversion µ(n) 1 π(x) = ∑ Π(x n ). n>0 n Furthermore Π(x + e) + Π(x − e) Z ∞ 1 dt = ( ) − ( ρ) + − lim Li x ∑ Li x 2 log 2. e→0 2 ρ x t − 1 t log t The integral logarithm function admits the following expansion x k! Li(x) ∼ ∑ ( )k log x k≥0 log x The prime number theorem can be stated as π(x) ∼ Li(x). In 1899, La Vallee´ Poussin proved that  √  π(x) ∼ Li(x) + O xe−a log x A precise statement can be given for H. von Koch showed in 1901 that, if and only if the Riemann hypothesis is true, the error term in the above relation can be improved to √ π(x) = Li(x) + O x log x . Another explicit formula, perhaps more tractable, is given connected to the logarithmic derivative of the Riemann zeta function. ζ0(s) ∞ Λ(n) Z(s) = − = ( ) ∑ s ζ s n=1 n where the Λ is given by  log p if n = pk for some prime p and k ≥ 1  Λ(n) = (5.2)  0 otherwise. It is neither multiplicative nor additive. We have by the Mobius¨ inversion formula: log n = ∑ Λ(d), Λ(n) = ∑ µ(d) log d. d|n d|n We define the second ψ(x) = ∑ Λ(n) = ∑ log p n≤x pν≤x and ( ψ(x + e) + ψ(x − e) ψ(x), x 6= pν ψ0(x) = lim = e→0 1 ν 2 ψ(x) − 2 log p, x = p . HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 135

The second explicit formula is [17](ch.5, th. 5.9): ρ 1 −2 x ψ0(x) = x + Z(0) − log(1 − x ) − lim . 2 T→+∞ ∑ ρ |=ρ|

5.2. According to Weil. Definition 5.1. A function F(x) is residual if the three following conditions are fulfilled (1) It is defined and differentiable for 0 < x < ∞ and for some γ > 0, we have F(x) = O(x−γ) as x tends to 0 and F(x) = O(xγ) as x tends to +∞, so that we may introduce for complex s the Mellin transforms Z 1 Z +∞ s−1 s−1 ξr(s) = F(x)x dx, ξl(s) = F(x)x dx 0 1

ξr is defined in a right half plane and ξl in a left half plane. (2) These two functions can be continued into one another in a domain D which is the exterior of a bounded set S. (3) If we denote the joint value of the continued function by ξ0(s), s = σ + iτ, then we have

lim ξ0(σ + iτ) = 0 |τ|7→+∞

uniformly on finite intervals σ1 ≤ σ ≤ σ2. This definition is motivated by Hecke’s theory on modular forms [57]. The following theorem of Bochner [7] gives a characterization Theorem 5.2. A function F(x) defined for 0 < x < ∞ is residual if and only if it can be represented as a series  n cn 1 1 F(x) = ∑ log , γ0 = lim sup |cn| n < ∞. n≥0 n! x c c ( ) ( ) = n ( −y) = n n The function ξ0 s is the Borel transform ξ0 s ∑ n+1 of the entire function of exponential type F e ∑ y . n≥0 s n≥0 n! We slightly modify the previous definition. Consider the class W of complex-valued functions f (x) on the positive half-line R+, continuous and continuously differentiable except for finitely many points at which both f (x) and f 0(x) have at most a discontinuity of the first kind, and at which the value of f (x) and f 0(x) are defined as the average of the right and left limits. Suppose also that there is δ > 0 such that f (x) = O(xδ) as x → 0+ and f (x) = O(x−1−δ) as x → +∞. Let f (s) be the Z ∞ dx f˜(s) = f (x)xs . 0 x This function is analytic in the strip −δ <

The first sum ranges over all nontrivial zeros of ζ(s) and is understood as

lim f˜(ρ), T→+∞ ∑ |=ρ|

5.3. According to Guinand. In [26] Guinand obtained under the Riemann hypothesis a general for- mula connecting prime numbers and non-trivial zeros of the Riemann zeta-functiom. His result can be stated as follows:

Theorem 5.4.   3t !  log p Z T t  1 Z ∞ 1 e 2 lim f (m log p) − f (t)e 2 dt − f (t) − dt T→∞ ∑ 1 0

( ) 1 1 Z T t −(2π) 2 lim g(ρ) − g(t) log dt , T→ ∑ ∞ 0<ρ

5.4. According to Bost. If we denote the characteristic function of prime numbers by ν(n), then [20]: ν(n) ω(n) ( ) = ( ) ( ) = P s ∑ s , P s ζ s ∑ s n≥1 n n≥1 n where ω(n) is the number of different prime factors in n ω(n) = ∑ ν(d) d|n which, by the Mobius¨ inversion formula, gives n ν(n) = µ(d)ω( ). ∑ d d|n

Remark 5.1. It is proved in [24] that the primes contain arbitrarily long arithmetic progressions and if 1 µ = ∑ δlog p, p∈P p then µ is a positive measure whose Fourier transform µˆ coincide with the distribution defined by the locally integrable function l(t) = P(1 + 2iπt), that is for every infinitely differentiable on the real line with compact support φ, we have [9] Z 1 φ(t)P(1 + 2iπt) dt = ∑ φˆ(log p). R p∈P p HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 137

5.5. Cramer’s Dirichlet series. There exist many versions of explicit formulas in which summation is taken over half of the zeroes of the Riemann zeta function [55] (p. 320), [40], using the functions

eiρz eiρz , , 0. ∑ ∑ 0( ) =ρ>0 ρ =ρ>0 ρζ ρ

For example

Theorem 5.5. If V(w) = ∑ eiρw =ρ>0 This series converges absolutely for =w > 0 and (1)V (w) admits a meromorphic continuation to the whole plane C cut from 0 to −∞, and has there first order poles at w = ± log pm with respective principal parts log p 1 log p 1 , 2iπ w − log pm 2iπ w + log pm (2) Near w = 0, we have ! 1 log w γ + log 2π − iπ 3 V(w) = + 2 + + ηi + wW(w) 2iπ 1 − e−iw w 4

where η > 0 and W(w) is single valued and regular for |w| < log 2.

(3) The function

1 1 log w − 2 iw U(w) = e V(iw) + w 4π sin 2 admits a single valued analytic continuation satisfying the functional equation w 1 U(w) + U(−w) = 2 cos − w 2 4 cos 2

5.6. Prime numbers theorem as a Tauberian theorem. We follow here the presentation of N.Wiener [58]. In general the singularities or the behaviors of a Dirichlet series

∞ λns f (s) = ∑ ane , λ0 < λ1 < ... n=0 depend on the coefficients (an) and the exponents (λn). However there are circumstances where only the ex- ponents determine the behavior of the sum or the singularities of the series on the boundary of the domain of convergence. An example of a such situation is given by the outstanding Tauberian theorem of Hardy and Littelwood [28] (Theorem 114, p.173):

∞ nk Theorem 5.6. (High-Indices Theorem) If f (x) = ∑k=0 akx converges for 0 < x < 1 where the exponents nk are positive integers satisfying the lacunarity condition n k+1 ≥ q > 1 nk − ∞ and if f (x) → A as x → 1 , then the infinite series ∑k=0 ak converges and its sum is A. 138 A. SEBBAR

5.7. Ramanujan Differential system. The divisor functions that we are going to define constitute an excellent laboratory where the Fourier analysis, arithmetic and spectral theory combine in a harmonious and fertile mixture We define for s ∈ C s σs(n) = ∑ d . d|n

The generating functions of the divisor sums σs(n) satisfy various identities, usually proved by making extensive use of the fact that they are modular forms. These generating functions, in case s = 1, 2, 3 are the Eisenstein series

2iπnτ E2(τ) = 1 − 24 ∑ σ1(n)e , =τ > 0 n≥1 together with ∞ 2iπnz E4(z) = 1 + 240 ∑ σ3(n), e , n=1 and 2iπnz E6(z) = 1 − 504 ∑ σ5(n) e n≥1 for which we have the following important Ramanujan differential system [48], [46]:  1 d 1 2  E2(z) = (E − E4).  2πi dz 12 2    1 d 1 E4(z) = 3 (E2E4 − E6)  2πi dz    1 d  E (z) = 1 (E E − E2), 2πi dz 6 2 2 6 4

Furthermore the Eisenstein series y = E2(z) satisfies de Chazy equation [59] 3 1 dy D3y − yD2y + (Dy)2 = 0, Dy = . (5.3) 2 2iπ dz The Chazy equation is an example of an ordinary differential equation with the general solution having moving natural boundary (Painleve´ property) and SL2(C)-invariance [12]. It also arose as a reduction of the self-dual Yang-Mills equation in [2]. It is remarkable that so many arithmetical relations between the divisor functions are consequences of the the Ramanujan differential system or the Chazy equation. For example it is readily seen from (5.3) that we have the following recursion relation for the sums of divisors of natural numbers σ1(n) = ∑ d: d|n

12 n−1 σ (n) = k(3n − 5k)σ (k)σ (n − k). 1 2( − ) ∑ 1 1 n n 1 k=1 a b The series E (z) is a holomorphic function on H = {z ∈ C, =z > 0}, the upper half plane, and if γ = ∈ 2 c d SL2(Z), we have  az + b  6c E = (cz + d)2 E (z) + (cz + d), (5.4) 2 cz + d 2 iπ that is to say that E2 is a quasi- of weight 2. Meanwhile, if we define 3 E∗(z) = E (z) − , y = =z , (5.5) 2 2 πy HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 139

∗ then E2 is a (not holomorphic) modular form of weight 2 for SL2(Z). Moreover, E2 is the logarithmic derivative of the discriminant function ∆(z), the classical cusp form of weight 12 for SL2(Z) ∞ ∆(z) = η(z)24 = q ∏ (1 − qn)24 , q = e2πiz. n=1 In fact, we have 1 ∆0(z) E (z) = . (5.6) 2 2iπ ∆(z) Ingham (1930) proved the prime number theorem using the particularly elegant identity of Ramanujan: ∞ σ (n)σ (n) ζ(s)ζ(s − a)ζ(s − b)ζ(s − a − b) a b = . ∑ s ( − − ) n=1 n ζ 2s a b More generally we define ∞ k+1 2iπnz E2k(z) = 1 + γ2k ∑ (−1) σ2k−1(n)e , (5.7) k=1 with k 4k γ2k = (−1) . B2k

For k > 1, the Eisenstein series E2k(z) can be written in more analytic form

1 −2k E2k(z) = ∑ (cz + d) (5.8) 2 c,d∈Z (c,d)=1

The series converges absolutely and uniformly on compact sets of H if k > 1, and E2k is a modular form of weight 2k, that is 2k E2k(γz) = (cz + d) E2k(z) a b for every γ = ∈ Γ = SL (Z). The sum over (c, d) = 1 is the same as the sum over γ ∈ Γ \Γ, where c d 2 ∞  1 n  Γ = ± , n ∈ Z . ∞ 0 1 The case of k = 1 is of a particular interest. It gives rise to the classical non-holomorphic Eisenstein series. Let 1 ys 1 ( ) = = =( )s E z, s ∑ 2s ∑ γz (5.9) 2 ∈Z |cz + d| 2 c,d γ∈Γ∞\Γ (c,d)=1 The normalized Eisenstein series is 1 1 ys E?(z, s) = π−sΓ(s)ζ(2s)E(z, s) = π−sΓ(s) . 2 2 ∑ |mz + n|2s (m,n)6=(0,0)  ∂2 ∂2  This series converges for 1 and it is an eigenfunction of the hyperbolic Laplacian ∆ = −y2 + . ∂x2 ∂y2 More precisely ∆E(z, s) = s(s − 1)E(z, s). Moreover E?(z, s) has the Fourier expansion

? −s s s−1 1−s s− 1 √ 2iπnx E (z, s) = π Γ(s)ζ(2s)y + π Γ(1 − s)ζ(2 − 2s)y + 2 |n| 2 σ1−2s(|n|) yK 1 (2π|n|y)e . ∑ s− 2 n6=0 140 A. SEBBAR

Z ∞ x cosh(t) where Ks(y) is the K-Bessel function, defined for 0 by Ks(x) = e cosh(st) dt. The Prime Number 0 Theorem is equivalent to the non-vanishing of the Riemann zeta-function on the line <(s) = 1 (Hadamard et de la Vallee´ -Poussin). Eisenstein series can be used to show this fact. Indeed the n-th Fourier coefficient of E(z, s) is 1 √ s− 2 Z 1 2|n| σ1−2s(|n|) yKs− 1 (2π|n|y) ( ) 2iπnx = 2 E z, s e dx −s . 0 π Γ(s)ζ(2s) Now the holomorphy of E(z, s) on <(s) = 1 implies the non-vanishing of ζ(s) on <(s) = 1. Namely, the information of Eisenstein series give rise to the information on ζ function. This is the beginning observation of Langlands-Shahidi method.

5.8. Robin’s theorem, Lagarias theorem. The behaviour of the sigma function σ1(n) is irregular. The asymptotic growth rate of the sigma function can be expressed by: σ (n) lim 1 = eγ. n→∞ log log n This result is Grnwall’s theorem, published in 1913 (Gronwall¨ 1913). His proof uses Mertens’ 3rd theorem, which says that 1 p lim = eγ. n→∞ ∏ − n p≤n, p prime p 1 In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality: γ σ1(n) ≤ e n log log n holds for all sufficiently large n (Ramanujan 1917). The largest known value that violates the inequality is n = 7! = 5040. In 1984 Guy Robin proves [50] Theorem 5.7. The previous inequality is true for all n > 5040 if and only if the Riemann hypothesis is true. Moreover Robin also proved, unconditionally, that the inequality 0.6483n σ (n) ≤ eγn log log n + 1 log log n holds for all n ≥ 3. Finally Jeffrey Lagarias showed in 2002 that the Riemann hypothesis is equivalent to the statement that Hn σ1(n) < Hn + ln(Hn)e for every natural number n > 1, where Hn is the n-th .

5.9. The Euler constant γ and the . There are so many properties in [30] on the Euler constant. n We would like to add some others. The function χ(z) = ∑ z2 is related to the paper folding problem [51], [52] n=0 and [53], where a relation to Jacobi theta function is given. Proposition 5.1. We have the equality Z 1 χ(x) γ = 1 + log 2 − dx 0 1 + x To see this we first repeat Vacca formula(1910) which says that:  1 1 1  γ = lim 1 + + + ··· + n→∞ 2 3 n 1 1  1 1 1 1   1 1 1  = − + 2 − + − + 3 − + · · · − + ··· 2 3 4 5 6 7 8 9 15 HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 141

In fact we know that for the Mercator’s series π 1 1 1 = 1 − + − + ··· 4 2 3 4 and we define 1 1 1 σ = 1 − + − · · · + − log 2. n 2 3 2n − 1 Then 1 1 1 σ + σ + ··· + σ = 1 + + + ··· + − n log 2 1 2 n 2 3 2n − 1 and therefore γ = lim (σ1 + σ2 + ··· + σn) n→∞ To finish the proof of Vacca’s formula we observe that

σn = τn + τn+1 + ··· where 1 1 1 1 τ = − + − · · · − . n 2n 2n + 1 2n + 2 2n+ − 1 Let F(x) = χ(x) − 1, since Vacca’s formula may be written in the form Z 1  x − x3 x3 − x7 x7 − x15  Z 1 F(x) γ = + 2 + 3 + ··· dx = dx 0 1 + x 1 + x 1 + x 0 x(1 + x) and since Z 1 F(x) 1 = = dx ∑ n 1 0 x n≥1 2 we obtain the desired result.

Another arithmetical function also related to the Euler gamma constant is the Euler totient function 1 Φ(n) = ]{j ∈ Z+, gcd(j, n) = 1} = n (1 − ) ∏ p p|n for which we have n = ∑ Φ(d). d|n Landau(1903) proved that log log n lim Φ(n) = e−γ n→∞ n

6. Almost periodicity

We refer to [34] Let wj(n) be 1 or 0 according as j is or is not a divisor of n, and let γ(n) be defined by: γ(n) γ(n)+ 2 | n, 2 1 - n.

Let r4(n) be the number of different representations of n as a sum of 4 squares. In other words

∞ ∞ !4 n n2 1 + ∑ r4(n)z = 1 + ∑ z , |z| < 1. n=1 n=1

Jacobi’s well known theorem concerning the representation of r4(n) in terms of σ1(n) may be written as follows: ∞ r (n) 1 + 2ω (n) σ (n) 1 + 2ω (n) ωj(n) 4 = 8 2 1 = 8 2 γ(n)+1 γ(n)+1 ∑ n 2 − 1 n 2 − 1 j=1 j 142 A. SEBBAR

If f (n) is a function defined for n = 0, 1, 2, ··· we define the average of f by 1 n M( f ) = lim f (j). n→∞ ∑ n j=1 Then we have [34] N !2 σ (n) ωj(n) M 1 − ∑ −→ 0, N → ∞. n j=1 j N ωj(n) Since the finite sums ∑ , N ≥ 1 are periodic, we then obtain: j=1 j σ (n) Theorem 6.1. The arithmetical function 1 is almost periodic (B2), that is almost periodic is in the sense of Besicovitch. n

Dirichlet showed that the average order of the σ0(n) = d(n) satisfies the following equality, ≥ valid for all x 1: √ ∑ d(n) = x log x + (2γ − 1)x + O( x) n≤x For prime divisors we have a nice probabilistic result: Theorem 6.2. (Erdos-Kac¨ Theorem). For any positive integer n ≥ 1, let ω(n) denote the number of prime divisors of n, counted without multiplicity. Then for any real numbers a < b, we have ( ) ( ) − Z b 2 1 ω n log log N 1 −x /2 lim 1 ≤ n ≤ N, a ≤ p ≤ b = e dx N→∞ N log log N 2π a

7. Motivation: Classical Polylogarithm function

The present section was the main origin of these notes. In [6], Besicovich solved the following problem: does there exist a continuous function f (x) in the interval 0 ≤ x ≤ 1 such that: (1) f (x) + f (1 − x) 6≡ 0, m (2) For each integer n > 0, ∑ f ( ) = 0. 1≤m≤n n Following an idea of Chowla and Bateman, we give an example of a non-trivial real-valued continuous function on the real line which has period unity, is not odd, and has the property  h  ∑ f = 0. 0 1, v 6= 0, −1, −2, ··· . (7.2) n=0 HARMONIC NUMBERS, HARMONIC SERIES AND ZETA FUNCTION 143 and to classical polylogarithm function with complex argument for v = 1 ∞ −s n Lis(z) = ∑ n z . (7.3) n=1 We also introduce ∞ −s 2iπnθ lis(z) = ∑ n e , θ ∈ R. (7.4) n=1 The function (7.1) has the remarquable Lindelof¨ decomposition: − Γ(1 − s)  1 s 1 ∞ (log z)r ( ) = + −v ( − ) Φ z, s, v v log z ∑ ζ s r, v , (7.5) z z r=0 r! | log z| < 2π, s 6= 1, 2, 3 ··· , v 6= 0, −1, −2, ··· . This expansion is a consequence of the Hurwitz’ formula [39]: ∞ 1 ζ(s, v) = 2(2π)−(s−1)Γ(1 − s) ∑ ns−1 sin(2πnv + πs),