Abstract
We investigate the behavior of the real part of the logarithmic derivatives of the Selberg zeta-functions ZPSL(2,Z)(s) and ZC (s) in the critical strip 0 < σ < 1. The functions ZPSL(2,Z)(s) and ZC (s) are defined on the modular group and on the compact Riemann surface, respectively.
Highlights
Let s = σ +it denote a complex variable
We start with the definition and some properties of the Riemann zeta-function
For σ > 1, the Riemann zeta-function is given by the series
Summary
The following reformulation for the Riemann hypothesis that all non-trivial zeros of ζ(s) lie on the line σ = 1/2 was given. As the Riemann zeta-function, the Selberg zeta-function ZPSL(2,Z)(s) has a meromorfic continuation to the whole complex plane, and satisfies the symmetric functional equation [8]. If the Riemann hypotesis is true for ζ(s), for a sufficiently large fixed t, the function |ZPSL(2,Z)(σ + it)| is decreasing for 0 < σ < 1/4 with respect to σ. Non-trivial zeros on the critical line σ = 1/2 and possibly, on the interval (0, 1) of the real axis, see [4, §2.4, Theorem 2.4.11] and [13] In this sense, the analogue of the Riemann hypothesis holds for ZC(s).
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