# Riemann-Type Functional Equations -- Julia Line and Counting Formulae --

Source: [Riemann-Type Functional Equations -- Julia Line and Counting Formulae --](<https://www.sota2.com/research/paper/68fa5c191e209131ce9ce20c>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T17:26:27.186Z

Authors: Athanasios Sourmelidis, J\\"orn Steuding, Ade Irma Suriajaya

Published: 2020-11-21T01:27:43.000Z

## 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 $a\\neq0$ and a function from the Selberg class $\\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $\\Delta$-factor of the functional equation of $\\mathcal{L}$ 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 $\\mathcal{L}(s)$ taken at these points.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2011.10692>)

[PDF](<https://arxiv.org/pdf/2011.10692.pdf>)