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On the Periodic Hurwitz Zeta-Function. a Javtokas, a Laurinčikas

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On the Periodic Hurwitz Zeta-Function. a Javtokas, a Laurinčikas On the periodic Hurwitz zeta-function. A Javtokas, A Laurinčikas To cite this version: A Javtokas, A Laurinčikas. On the periodic Hurwitz zeta-function.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2006, 29, pp.18 - 36. hal-01111465 HAL Id: hal-01111465 https://hal.archives-ouvertes.fr/hal-01111465 Submitted on 30 Jan 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. Hardy-Ramanujan Journal Vol.29 (2006) 18-36 On the periodic Hurwitz zeta-function A. Javtokas, A. Laurinˇcikas Abstract.In the paper an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved. 1. Introduction Let N, N0, R and C denote the sets of all positive integers, non-negative integers, real and complex numbers, respectively, and let a = a ; m Z f m 2 g be a periodic sequence of complex numbers with period k 1. Denote by s = σ + it a complex variable. The periodic zeta-function ζ(s≥; a), for σ > 1, is defined by 1 a ζ(s; a) = m ; ms m=1 X and by analytic continuation elsewhere. Define 1 k−1 a = a : k m m=0 X A. Javtokas, A. Laurinˇcikas 19 If a = 0, then ζ(s; a) is an entire function, while if a = 0, then the point s = 1 6 is a simple pole of ζ(s; a) with residue a. The function ζ(s; a) satisfies the functional equation [22] k s Γ(s) πis πis ζ(1 s; a ) = exp ζ(s; a ) + exp ζ(s; a ) ; − 2π p 2 ∓ − 2 k where Γ(s) is the Euler gamma-function, and 1 k 2πilm a = al exp : m Z : (pk k 2 ) Xl=1 The function ζ(s; a) was studied by many authors. In [3] the function ζ(s; a) appears as a special case of the periodic Lerch zeta-function with its functional equation. The papers [7] and [22] are devoted to Hamburger-type theorems for ζ(s; a). In [6] the Kronecker limit formula for ζ(s; a) is obtained. The mean square of ζ(s; a) is studied in [11], [18] and [19]. Probabilistic limit theorems in various spaces are proven in [10] and [12]. The paper [9] contains some expansions for ζ(s; a) and its derivatives as well as a Voronoi type formula for dl(m) = ad1 : : : adl ; d1:::Xdl=m which is obtained by using the properties of ζ(s; a). The zero-distribution of ζ(s; a) is investigated in [24]. The zero-free regions and formulas for the number of non-trivial zeroes of ζ(s; a) are established. Moreover, in [25] an important result on the universality of ζ(s; a) is obtained. Let meas A denote the Lebesgue measure of the set A R, and let, for T > 0, f g ⊂ 1 ν (: : :) = meas τ [0; T ] : : : : ; T T f 2 g where in place of dots a condition satisfied by τ is to be written. Suppose that k is an odd prime, am is not a multiple of a character modulo k, and ak = 0. 20 On the periodic Hurwitz zeta-function Let K be a compact subset of the strip s C : 1=2 < σ < 1 with connected f 2 g complement, and let f(s) be a continuous function on K which is analytic in the interior of K. Then in [25] it is proved that, for any " > 0, lim inf νT sup ζ(s + iτ; a) f(s) < " > 0: T !1 j − j s2K Note that the universality of the Riemann zeta-function was discovered by S.M. Voronin [26]. Later, the Voronin theorem was improved and generalized by many authors, see, the survey papers [8], [16] and [21]. In this paper we consider a generalization of the function ζ(s; a). Let 0 < α 1, and, for σ > 1, ≤ 1 a ζ(s; α; a) = m : (m + α)s m=0 X For σ > 1, we have k−1 1 1 k−1 1 1 ζ(s; α; a) = a = a l (m + α)s l (l + rk + α)s l=0 r=0 l=0 r=0 X mX=l+rk X X (1) 1 k−1 1 1 1 k−1 l + α = a = a ζ s; ; ks l l+α s ks l k r=0 r + k Xl=0 X Xl=0 where ζ(s; α) is the Hurwitz zeta-function. Therefore, the function ζ(s; α; a) is a linear combination of the Hurwitz zeta-functions, and equality (1) gives analytic continuation of the function ζ(s; α; a) to the whole complex plane, where it is regular, except, maybe, for a simple pole at s = 1 with residue a (if a = 0). 6 If am = 1 and k = 1, the function ζ(a; α; a) reduces to the Hurwitz zeta-functionf g ζf(s;gα). If additionally α = 1, then ζ(s; α; a) becomes the Rie- 2πi lm mann zeta-function. The sequence a = e k ; m N , (l; k) = 1, clearly, l 2 0 n o A. Javtokas, A. Laurinˇcikas 21 is periodic with period k. Therefore, in the case a = al the function ζ(s; α; a) reduces to the Lerch zeta-function 1 e2πiλm L(λ, α; s) = ; σ > 1; (m + α)s m=0 X with rational parameter λ. Thus, the function ζ(s; α; a) is a generalization of classical zeta-functions, and it is reasonable to call ζ(s; α; a) either the periodic Hurwitz zeta-function, or the Hurwitz zeta-function with periodic coefficients. The function ζ(s; α; a) has been investigated in [2], true with a small differ- ence in the definition of ζ(s; α; a). Indeed, in our notation, in [2] the function a F (s) = ζ(s; α; a) 0 − αs has been considered. Applying their original approximate functional equation for the function ζ(s; α) α−s, the authors obtained in [2] the following inter- esting estimate. Suppose− that a = 0, max a A and H = T 1=3. Then, for j j jj ≤ T 2, ≥ 1 T +H F (1=2 + it) 2 A2k log3 T + A2k log k H j j ZT uniformly in α. Also, in [2] a mean-square estimate for twists of F (s) with some sequence has been obtained. The aim of this note is to prove the universality of the function ζ(s; α; a) for some values of α and certain sequence a. Theorem 1. Suppose that α is a transcendental number and min0≤m≤k−1 am > 0. Let K be a compact subset of the strip s C : 1=2 < σ < 1 j withj connected complement, and let f(s) be a continuousf 2 function on K whichg is analytic in the interior of K. Then, for any " > 0, 22 On the periodic Hurwitz zeta-function lim inf sup ζ(s + iτ; α; a) f(s) < " > 0: T !1 j − j s2K Note that the universality theorem for the Hurwitz zeta-function with tran- scendental parameter was first proved in the Ph.D. thesis of B. Bagchi [1]. He proposed a new method for the proof of universality for Dirichlet series based on limit theorems in the sense of weak convergence of probability measures in the space of analytic functions. We will apply this method of limit theorems as well as other tools used in [1] for the proof of Theorem 1. 2. The mean square of ζ(s; α; a) As it was mentioned above, for the proof of Theorem 1 we need a limit theorem in the sense of week convergence of probability measures on the space of analytic functions for the function ζ(s; α; a). The proof of theorem of such a kind is based on the mean square estimate for ζ(s; α; a) in the region σ > 1=2. Theorem 2. Let σ > 1=2. Then 1 T ζ(σ + it; α; a) 2dt 1: T j j σ,α,a Z0 Proof. Suppose that max a C. Then in view of (1) we find that 0≤j≤k−1 j jj ≤ k−1 l + α 2 ζ(s; α; a) 2 k−2σ2k−1 a 2 ζ s; j j ≤ j lj k l=0 X (2) k−1 l + α 2 k−2σ2k−1C2 ζ s; : ≤ k l=0 X By Theorems 3.3.1 and 3.3.2 from [16], for σ > 1=2, A. Javtokas, A. Laurinˇcikas 23 T ζ(σ + it; α) 2dt T: j j σ,α Z1 Therefore, taking into account (2), we obtain the theorem. 3. A limit theorem Let D = s C : 1=2 < σ < 1 . Denote by H(D) the space of analytic on D functions equippf 2 ed with the topgology of uniform convergence on compacta. Let (S) stand for the class of Borel sets of the space S. B Define 1 Ω = γm; m=0 Y where γm is the unit circle γ = s C : s = 1 for every m N0. With the product topology and pointwisefm2ultiplicationj j gthe infinite-dimensional2 torus Ω is a compact topological Abelian group.
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