Zeta Regularized Trigonometric Products Over Zeros Of The Riemann Zeta Function
Efe Gürel
TL;DR
The paper develops a zeta-regularized framework for products over the nontrivial zeros of the Riemann zeta function, focusing on trigonometric and exponential factors of the form $\sin(\alpha\rho - z)$ and $e^{-i(\alpha\rho - z)} - \omega$. Under suitable arg-regularity conditions, it proves the regularizability of the associated zeta functions and derives explicit closed forms for the regularized products in terms of a quadratic regulator $F$ and zeta-based Pochhammer symbols: $S(\underline{z};\underline{\alpha}) = e^{-F(\underline{z};\underline{\alpha})}\prod_{k=1}^{n}(e^{-2iz_k};e^{-2i\alpha_k})_{\zeta}$, and its variant with $\omega_k$: $\tilde S(\underline{z};\underline{\alpha},\underline{\omega}) = e^{-\tilde F(\underline{z};\underline{\alpha})}\prod_{k=1}^{n}(\omega_k e^{-iz_k};e^{-i\alpha_k})_{\zeta}$. The multiplicative anomaly (discrepancy) is captured by $F=\sum_k F(z_k;\alpha_k) - F(\underline{z};\underline{\alpha})$ (and analogously for $\tilde F$). The results are obtained via binomial expansions and relationships to Cramér-type functions, with regularization conditions ensuring consistent definitions across different zeta-function representations. The work also discusses connections to weak forms of the Riemann hypothesis through magnitudes of the trigonometric products and proposes extensions to Selberg zeta-type settings.
Abstract
We prove a novel zeta regularized product formula concerning regularization of trigonometric products over non-trivial zeros of the Riemann zeta function. Furthermore, we calculate the discrepancies of such regularized products. In special cases, our formula reduces to the Kimoto-Wakayama formula. A conjectural relationship between such products and a weak Riemann hypothesis is speculated.