Related papers: The $\zeta$-regularized product over all primes
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…
We rewrite Riemann Zeta function as a sum over the primes. Each term of the sum is a product that depends only on the summation index (a prime) and the primes following it.
We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…
In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $\zeta(s)$ valid for $\Re(s)>1$, as well as similar formulas for $\zeta(s)$ valid for an even and odd $k$th positive integer argument.…
In this paper, we study the closed points of arithmetic schemes. We accomplish this by showing that the product of the cardinals of residue fields of closed points in an arithmetic scheme can be regularized. This regularization yields a new…
In this paper, the problem of multiplicative anomaly of zeta regularization is solved for polynomials. For a regularizable sequence $\Lambda$, we explicitly calculate the zeta regularized product of $(\Lambda-z_1)\dots(\Lambda-z_n)$ for…
We provide heuristic computations \`a la Euler of the regularized infinite products of Gauss and Eisenstein integers and primes. Our approach, yielding explicit expressions, is inspired by the work by Mu\~noz Garc\'ia and P\'erez-Marco, who…
In this paper we consider a family of multiple Hurwitz zeta values with bi-indices parameterized by $\mu$ with $\Ree(\mu)>0$. These values are equipped with both the $\mu$-stuffle product from their series definition and the shuffle product…
As a generalization of [KMW], we introduce a higher Riemann zeta function for an abstract sequence. Then we explicitly determine its regularized product expression.
We evaluate the multiple zeta values $\zeta(\{2\}^k)$ by proving a certain factorization property. The proof uses a combinatorial bijection and elementary telescoping series. We show how the infinite product for the sine function in fact…
The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of…
We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials…
What is the product of all {\em odious} integers, i.e., of all integers whose binary expansion contains an odd number of $1$'s? Or more precisely, how to define a product of these integers which is not infinite, but still has a "reasonable"…
We focus on a well-known convergence phenomenon, the fact that the $\zeta$ zeros are the universal singularities of certain Euler products.
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…
This analysis which uses new mathematical methods aims at proving the Riemann hypothesis and figuring out an approximate base for imaginary non-trivial zeros of zeta function at very large numbers, in order to determine the path that those…
In this paper we prove a regularized product expansion for the two-variable zeta functions of number fields introduced by van der Geer and Schoof. The proof is based on a general criterion for zeta-regularizability due to Illies. For number…
We introduce finite multiple zeta values of general level and discuss the relationship between the non-zeroness of these values and regular or non-Wieferich primes. Because it's challenging to prove the infinitude of these types of primes,…
We propose a regularization technique and apply it to the Euler product of zeta functions, mainly of the Riemann zeta function, to make unknown some clear. In this paper that is the first part of the trilogy, we try to demonstrate the…
We consider the problem whether the ordinates of the non-trivial zeros of $\zeta(s)$ are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros $(x_n)$ are uniformly distributed modulo 1. Odlyzko conjectured…