Regularized Euler product for the zeta function and the Birch and Swinnerton-Dyer and the Beilinson conjecture

TL;DR

This paper introduces a new regularization of the Euler product for the Riemann zeta function and extends the technique to elliptic curve L-functions, providing insights into the Birch and Swinnerton-Dyer and Beilinson conjectures.

Contribution

It proposes a novel regularization method for Euler products and applies it to elliptic curve L-functions to analyze their properties in the critical strip.

Findings

Regularized Euler product expression for the zeta function.

Application of the method to elliptic curve L-functions.

Potential implications for BSD and Beilinson conjectures.

Abstract

We present another expression to regularize the Euler product representation of the Riemann zeta function. % in this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a new one as the limit of the product of some terms derived from the usual Euler product. We also refer to the relation between the Bernoulli number and $P(z)$, which is an infinite summation of a $z$ power of the inverse primes. When we apply the same technique to the $L$-function associated to an elliptic curve, we can evaluate the power of the Taylor expansion for the function even in the critical strip, which is deeply related to problems known as the Birch and Swinnerton-Dyer conjecture and the Beilinson conjecture.