Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions

Matiyasevich, Yuri; Saidak, Filip; Zvengrowski, Peter

doi:10.4064/aa166-2-4

Cited by 11 publications

(14 citation statements)

References 11 publications

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“…Proof. As remarked in [MSZ14], for a homolorphic function f , a simple calculation, using the Cauchy-Riemann equation, leads to…”

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confidence: 98%

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

Friedli¹

2017

Tohoku Math. J. (2)

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We initiate the study of spectral zeta functions ζ X for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function ζ(s) is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of ζ(s). We relate ζ Z to Euler's beta integral and show how to complete it giving the functional equation ξ Z (1 − s) = ξ Z (s). This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions d we provide a meromorphic continuation of ζ Z d (s) to the whole plane and identify the poles. From our aymptotics several known special values of ζ(s) are derived as well as its non-vanishing on the line Re(s) = 1. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function F 1 via an Euler-type integral formula due to Picard.

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“…Proof. As remarked in [MSZ14], for a homolorphic function f , a simple calculation, using the Cauchy-Riemann equation, leads to…”

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confidence: 98%

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

Friedli¹

2017

Tohoku Math. J. (2)

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“…A number of interesting results on the horizontal monotonicity of various Dirichlet series, for instance, for the Riemann zeta function and related L‐functions, have been known, some of which can be attributed to Spira , Saidak and Zvengrowski , Matiyasevich, Saidak and Zvengrowski , Zhang , and the references therein. Concerning horizontal monotonicity along the real line, Alzer proved the monotonicity properties of a function related to the Riemann zeta function given by

{(() 1 - \frac{1}{ζ (s)})}^{1 / false(s - a false)}

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Monotonicity properties of ‐functions

Chaubey

Koutsaki

Zaharescu

2018

Mathematische Nachrichten

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We present some monotonicity results for a class of Dirichlet series generalizing previously known results. The fact that ζ′false(sfalse) is in that class presents a first example of an arithmetic function for which the associated Dirichlet series is completely monotonic, but not logarithmically completely monotonic. Lastly, we use similar techniques to prove another formulation of the Riemann hypothesis for the L‐function associated to the Ramanujan‐tau function.

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“…In the paper [11], it was proved the following relation between functions ζ(s) and ξ(s). Sondow and Dumitrescu proved in [17] the following theorem for the function ξ(s).…”

Section: Introductionmentioning

confidence: 99%

“…Later, Theorem 3 was reproved in [11] in a slightly different way. Related properties of the functions ζ(s) and ξ(s) in the critical strip were also investigated in [15].…”

Section: Introductionmentioning

confidence: 99%

“…Then |f (s)| is strictly decreasing with respect to σ in D, i.e., for each s 0 ∈ D, there exists δ > 0 such that |f (s)| is strictly monotonically decreasing with respect to σ on the horizontal interval from s 0 − δ to s 0 + δ. (b) Conversely, if |f (s)| is decreasing with respect to σ in D, then Re f (s) f (s) ≤ 0 for all s ∈ D such that f (s) = 0.The proof of the lemma is given in[11].Remark 1. Of course, the analogous results hold for monotonically increasing |f (s)| and Re f (s) f (s) > 0.…”

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confidence: 99%

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