The Epstein–Riemann conjecture for the identity Epstein zeta function

Let αN\alpha\in\mathbb{N}, let ζQ(s)\zeta_Q(s) be the Epstein zeta function associated with a real-valued positive definite α×α\alpha\times\alpha matrix QQ, and let ζid(s)\zeta_{\mathrm{id}}(s) denote the case Q=idQ=\mathrm{id}. A critical zero is a zero sCs\in\mathbb{C} with 0<Re(s)<α/20<\mathrm{Re}(s)<\alpha/2. Epstein–Riemann conjecture. All critical zeros sCs\in\mathbb{C} of ζid(s)\zeta_{\mathrm{id}}(s) have real part

Re(s)=α/4.\mathrm{Re}(s)=\alpha/4.

This extends the Riemann-hypothesis phenomenon to Epstein zeta functions and, via the identity ζid(s)=ζ(Δ,s)\zeta_{\mathrm{id}}(s)=\zeta(\Delta,s), to the spectral zeta function of a flat torus. The source presents the assertion as a conjecture and gives no resolution status.

Sources & referencesView supporting material

Primary source

Alexander Meiners and Boris Vertman, “Spectral zeta function on discrete tori and Epstein-Riemann conjecture”, arXiv:2202.02420 (2022).

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