ON SOME SERIES FORMED BY VALUES OF THE RIEMANN ZETA FUNCTION Claude Henri Picard To cite this version: Claude Henri Picard. ON SOME SERIES FORMED BY VALUES OF THE RIEMANN ZETA FUNCTION. 2015. hal-01228352 HAL Id: hal-01228352 https://hal.archives-ouvertes.fr/hal-01228352 Preprint submitted on 13 Nov 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON SOME SERIES FORMED BY VALUES OF THE RIEMANN ZETA FUNCTION CLAUDE HENRI PICARD Abstract. The partial fraction expansion of coth(πz), due to Euler, is gener- alized to power series having for coefficients the Riemann zeta function evalu- ated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet series is also proposed. The resulting formulas are new, as far as we know, since they could not be found in any of the classical or recent handbooks of formulas that were at our disposal. 1. Origin of the series The starting point for this paper is the known formula [10, p. 49]: 2 ∞ π 1 if z =0, = 6 n2 + z2 π 1 n=1 coth(πz) if 0 < z < 1. X 2z − 2z2 | | These two values must obviously be connected. Indeed, from the Taylor series for z = 0, we obtain the expansion ∞ 1 π2 π4 π6 = z2 + z4 + O(z6), (z 0), n2 + z2 6 − 90 945 → n=1 X where the classical values of the Riemann’s zeta function for the even integers can be recognized immediately. The same question can now be asked for similar partial fraction expansions such as ∞ ∞ 1 1 , ,... n3 + z3 n4 + z4 n=1 n=1 X X Our results presented below extend significantly some particular formulas pub- lished recently in [4] and [6] (although it was written before knowing these papers), notably by considering Ces`aro summation and more general Dirichlet series. 2. Formula of order p Theorem 1. For a complex z of modulus less or equal to 1, z = 1: ∞ ∞ 6 − 1 (1) = ( 1)kζ(2k + 2)zk, n2 + z − n=1 k X X=0 ∞ ∞ 1 (2) = ( 1)kζ(3k + 3)zk, n3 + z − n=1 k X X=0 and generally for p C such that e(p) > 1 (formula of order p) : For all complex numbers∈ z of modulusℜ less or equal to 1, z = 1: 6 − ∞ ∞ 1 (3) = ( 1)kζ(pk + p)zk. np + z − n=1 k X X=0 Date: 28 may 2015. 1 2 CLAUDE HENRI PICARD Proof. We prove only the formula of order p, since the first two formulas are special cases. Let us first assume z < r0 < 1. This condition implies the normal con- vergence1 of the series and hence| | the legitimate inversion of the summations [9, p. 104]. We have ∞ ∞ ∞ ∞ 1 1 1 1 z kp = = ( 1)k np + zp np 1 + (z/n)p np − n n=1 n=1 n=1 k X X X X=0 and we obtain, with the inversion of summations, ∞ ∞ ∞ ∞ 1 1 = ( 1)kzpk = ( 1)kζ(pk + p)zpk np + zp − np(k+1) − n=1 k n=1 k X X=0 X X=0 Substituting z to zp yields the formula of order p. We thus have an analytical function of z in D(0, r0) which can be extended to the whole open disc D(0, 1) because the convergence radius is 1 from the Cauchy-Hadamard formula. On the convergence circle, the analytical function ∞ k pk fp(z)= ( 1) ζ(pk + p)z − k X=0 has a pole at z = 1. From the formula, the only problematic term in the left hand side is the first term− whose denominator is zero if z = 1. We conclude that the − function fp(z) has a simple pole on the convergence circle z =1 at z = 1. | | − Theorem 2. The function fp(z) is meromorphic on C, and its singularities are an infinity of simple poles at z = np (n N∗) with residue 1. − ∈ Proof. All the singularities of fp must appear in the left hand side of the formula of order p. We begin by the pole at 1. We have to calculate the residue by the formula − lim (z + 1)fp(z). z→−1 From the formula of order p, we have ∞ 1 lim (z + 1)fp(z) = lim z→−1 z→−1 np + z n=1 X ∞ 1 1 = lim (z + 1) + lim (z + 1) z→−1 1+ z z→−1 np + z n=2 X =1+0=1. Hence this pole is simple and has residue 1. For the other poles, the same demon- stration can be given by considering np + z instead of 1+ z. Remark 1. When p is not real, the poles of fp(z) are on a logarithmic spiral. The poles are on a straight line (the real axis) only when p is real. It is the only ana- lytical function that we know of whose poles are on a spiral, without being specially constructed by using the Mittag-Leffler’s theorem on the principal parts. 1[2, p. 124] “Series wich satisfy the M-test have been called normally convergent by Baire.” ON SOME SERIES FORMED BY VALUES OF THE RIEMANN ZETA FUNCTION 3 3. First generalization Let m and p be two complex numbers such that e(m) < e(p) 1. We have, for z < 1, ℜ ℜ − | | ∞ ∞ ∞ ∞ nm 1 1 1 z kp = = ( 1)k np + zp np−m 1 + (z/n)p np−m − n n=1 n=1 n=1 k X X X X=0 and then, since the condition on m ensures the absolute convergence of the series, ∞ ∞ ∞ nm 1 = ( 1)kzpk , np + zp − np(k+1)−m n=1 k n=1 X X=0 X Hence Theorem 3. Let m and p be two complex numbers such that e(m) < e(p) 1. Then, for z 1, z = 1, ℜ ℜ − | |≤ 6 − ∞ ∞ nm (4) = ( 1)kζ(pk + p m)zk, np + z − − n=1 k X X=0 and the power series ∞ ( 1)kζ(pk + p m)zk − − k X=0 defines a meromorphic function which can be extended to the whole complex plane, except for an infinity of simple poles at z = np (n N∗) with residue nm. − ∈ 4. Second generalization Let m be an integer. For z < 1, we have | | ∞ ∞ ∞ ∞ lnm(n) lnm(n) 1 lnm(n) z kp = = ( 1)k np + zp np 1 + (z/n)p np − n n=1 n=1 n=1 k X X X X=0 and then ∞ ∞ ∞ lnm(n) lnm(n) = ( 1)kzpk . np + zp − np(k+1) n=1 k n=1 X X=0 X Hence Theorem 4. Let m an integer. For z 1, z = 1, | |≤ 6 − ∞ ∞ lnm(n) (5) = ( 1)m ( 1)kζ(m)(pk + p)zk, np + z − − n=1 k X X=0 and the power series ∞ ( 1)kζ(m)(pk + p)zk − k X=0 defines a meromorphic function which can be extended to the whole complex plane, except for an infinity of simple poles at z = np (n N∗) with residue ( 1)m lnm(n). − ∈ − By combining the two previous theorems, we get a more general result. Theorem 5. Let m be an integer, and p, q be two complex numbers such that e(q) < e(p) 1. ℜ For zℜ 1, −z = 1, | |≤ 6 − ∞ ∞ nq lnm(n) (6) = ( 1)m ( 1)kζ(m)(pk + p q)zk, np + z − − − n=1 k X X=0 4 CLAUDE HENRI PICARD and the power series ∞ ( 1)kζ(m)(pk + p q)zk − − k X=0 defines a meromorphic function which can be extended to the whole complex plane, except for an infinity of simple poles at z = np (n N∗) with residue ( 1)mnq lnm(n). − ∈ − 5. Third generalization The same principle works with more complex expressions. We have, with two terms, for z < 1/ max( a , b ), e(p) > 1, e(q) > 1, | | | | | | ℜ ℜ ∞ ∞ ∞ 1 (7) = ( 1)k+lζ(p + q + pk + lq)akblzk+l. (np + az)(nq + bz) − n=1 l k X X=0 X=0 With three terms, for z < 1/ max( a , b , c ), e(p) > 1, e(q) > 1, e(r) > 1, | | | | | | | | ℜ ℜ ℜ ∞ 1 (8) = (np + az)(nq + bz)(nr + cz) n=1 X ∞ ∞ ∞ ( 1)k+l+mζ(p + q + r + pk + lq + mr)akblcmzk+l+m. − m=0 l k X X=0 X=0 And so on. 6. General formulas As previously we work with expressions of the form an/(bn + z) with an/bn absolutely summable. We remark that these expressions, as function of z, are ∗ P P absolutely summable for z C bn n N . We have also ∈ −{ | ∈ } Lemma 1. (Abel, 1826) Let the entire series ∞ k f(z)= akz k X=0 of convergence radius 1. If ∞ ak k X=0 converge to L then, for z in an Stolz angle of vertex 1, ∞ k lim akz = L = f(1). z→1 k X=0 Theorem 6. Consider the Dirichlet’s series ∞ a f(s)= n ns n=1 X having σa =+ as abscissa of absolute convergence. Let s a complex number such 6 ∞ that e(s) > σa.