Regularizations of the Euler product representation for zeta functions and the Birch--Swinnerton-Dyer conjecture

TL;DR

This paper introduces a new regularization method for Euler product representations of zeta functions, which helps analyze zeros and relates to the Birch--Swinnerton-Dyer conjecture by examining associated L-functions.

Contribution

A novel regularization approach that avoids the Möbius function, enabling finite residuals and insights into zeros and the Birch--Swinnerton-Dyer conjecture.

Findings

Confirmed finite behavior of residual terms for the Riemann zeta function

Applied technique to L-functions of elliptic curves

Connected Taylor expansion at poles to the Birch--Swinnerton-Dyer conjecture

Abstract

We consider a variant expression to regularize the Euler product representation of the zeta functions, where we mainly apply to that of the Riemann zeta function in this paper. The regularization itself is identical to that of the zeta function of the summation expression, but the non-use of the M\"oebius function enable us to confirm a finite behavior of residual terms which means an absence of zeros except for the critical line. Same technique can be applied to the $L$-function associated to the elliptic curve, and we can deal with the Taylor expansion at the pole in critical strip which is deeply related to the Birch--Swinnerton-Dyer conjecture.