Spectral zeta function on discrete tori and Epstein-Riemann Hypothesis

Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

“…Let us remark that the Riemann Hypothesis can be reformulated solely in terms of a hypothetical functional symmetry of the standard type s ↔ 1 − s for ζ Z (s/2) and ζ Z/nZ (s/2). This is not an accidental fact, as it was shown to extend in [F16] and [MV22] to some cases of a generalized Riemann hypothesis.…”

“…Note that asymptotical symmetry of the type s vs. 1−s for ζ Z/nZ (s/2) and related finite sums are not a trivial matter, in fact certain versions of it are equivalent to various Riemann hypotheses: for ζ(s) as shown in [FK17], for certain Dirichlet L-functions as proven in [F16], and for the Dedekind zeta function of the Gaussian rationals as established in [MV22].…”

Transcendence and the Expression of the Spectral Series of a Class of Higher Order Differential Operators

“…is transcendental for m = 3, 4, 6. However, if the Mercer's Theorem and Green function are used, it is required that the series Recently, many papers [8], Wainger [20], Meiners and Vertman [21] study the special values of spectral zeta functions on the discrete tori. Recall that the spectral zeta function associated with the Cayley graph Z/NZ is (cf.…”