Riemann-type functional equations: Julia line and counting formulae

Athanasios Sourmelidis; Jörn Steuding; Ade Irma Suriajaya

doi:10.1016/j.indag.2022.08.002

Riemann-type functional equations: Julia line and counting formulae

Athanasios Sourmelidis^*, Jörn Steuding, Ade Irma Suriajaya

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a≠0 and a function from the Selberg class L, we prove a Riemann–von Mangoldt formula for the number of a-points of the Δ-factor of the functional equation of L and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these a-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L(s) taken at these points.

Original language	English
Pages (from-to)	1236-1262
Number of pages	27
Journal	Indagationes Mathematicae
Volume	33
Issue number	6
DOIs	https://doi.org/10.1016/j.indag.2022.08.002
Publication status	Published - Nov 2022

Keywords

a-points
Extended Selberg class
Functional equation
Landau formula
Riemann–von Mangoldt formula

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.indag.2022.08.002

Cite this

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title = "Riemann-type functional equations: Julia line and counting formulae",

abstract = "We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a≠0 and a function from the Selberg class L, we prove a Riemann–von Mangoldt formula for the number of a-points of the Δ-factor of the functional equation of L and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these a-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L(s) taken at these points.",

keywords = "a-points, Extended Selberg class, Functional equation, Landau formula, Riemann–von Mangoldt formula",

author = "Athanasios Sourmelidis and J{\"o}rn Steuding and Suriajaya, {Ade Irma}",

year = "2022",

month = nov,

doi = "10.1016/j.indag.2022.08.002",

language = "English",

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pages = "1236--1262",

journal = "Indagationes Mathematicae",

issn = "0019-3577",

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TY - JOUR

T1 - Riemann-type functional equations

T2 - Julia line and counting formulae

AU - Sourmelidis, Athanasios

AU - Steuding, Jörn

AU - Suriajaya, Ade Irma

PY - 2022/11

Y1 - 2022/11

N2 - We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a≠0 and a function from the Selberg class L, we prove a Riemann–von Mangoldt formula for the number of a-points of the Δ-factor of the functional equation of L and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these a-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L(s) taken at these points.

AB - We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a≠0 and a function from the Selberg class L, we prove a Riemann–von Mangoldt formula for the number of a-points of the Δ-factor of the functional equation of L and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these a-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L(s) taken at these points.

KW - a-points

KW - Extended Selberg class

KW - Functional equation

KW - Landau formula

KW - Riemann–von Mangoldt formula

UR - https://doi.org/10.1016/j.indag.2022.08.002

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U2 - 10.1016/j.indag.2022.08.002

DO - 10.1016/j.indag.2022.08.002

M3 - Article

SN - 0019-3577

VL - 33

SP - 1236

EP - 1262

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 6

ER -

Riemann-type functional equations: Julia line and counting formulae

Abstract

Keywords

ASJC Scopus subject areas

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