Related papers: Euler Product Sieve

In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets…

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.…

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…

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the…

We introduce a new set of prime numbers functions including an exact Generating Function and a Discriminating Function of Prime Numbers neither based on prime number tables nor on algorithms. Instead these functions are defined in terms of…

One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no…

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…

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…

In the present work we show that the Dirichlet series with the Euler product having analytical continuation to the critical strip without singularities, in some natural conditions, can be approximated by partial products of Euler type in…

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an…

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…

We have proposed a regularization technique and apply it to the Euler product of zeta functions in the part one. In this paper that is the second part of the trilogy, we give another evidence to demonstrate the Riemann hypotheses by using…

We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between…

In 1874, Mertens proved the approximate formula for partial Euler product for Riemann zeta function at $s=1$, which is called Mertens' theorem. In this paper, we generalize Mertens' theorem for Selberg class and show the prime number…

In this paper, we calculate the absolute tensor square of the Dirichlet $L$-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the…

In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of…

In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the…

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…

The first step in the formulation and study of the Riemann Hypothesis is the analytic continuation of the Riemann Zeta Function (RZF) in the full Complex Plane with a pole at $s=1$. In the current work, we study the analytic continuation of…

First idea is to compute a quantity like the angular momentum with respect to (0, 0), of an unitary mass of coordinates (<[Xi(s)], =[Xi(s)]) while =[s] is the time, and, <[s] = constant. If we impose that the derivative along <[s], at…