EULER PRODUCT ASYMPTOTICS FOR DIRICHLET -FUNCTIONS

The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.

Apuntes para la crítica del discurso progresista históricamente existente

From a dialectical hermeneutic methodology that discusses different ideological texts produced in particular political contexts of power, the objective of the article lies in defining a critical line that points out inconsistencies and contradictions before what generically can be defined without much conceptual precision as progressive discourse, that is: a discursive formation that brings together different social movements, civil organizations and political parties --in theory-- of frank counter-hegemonic character. It is concluded that morally the progressive discourse is not by itself bad, good, or neutral, everything will depend, ultimately, on the tastes and preferences of each person built by their cultural biases and, more specifically, in the heat of the processes of political socialization by which they have been conditioned ontologically. Therefore, rather than a critique of the historically existing progressive discourse, a review must be made of the tendentious way in which certain political actors use this discourse to validate their hegemonic interests in their context of action. In addition, in many verifiable respects this discourse can mean the justification of authoritarian practices that contravene the enjoyment and enjoyment of fundamental rights, which does not mean that the authors bet on conservative political and ideological positions.

The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$ , where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$ , whereas the previous best was $T^{1/3}$ , from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$ . Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $ , this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$ .

Riemann Hypothesis Holds virtually true

If infinity times zero equal zero then Zeta function has an analytic continuity over the whole complex plan except a simple pole at 1. Since infinity times zero is undefined, then Riemann' s approach remain not sharp. However, it is true that the non trivial zeros lie at the critical line x = 1/2 despite there simultaneous virtual existence or in another words, such zeros are assumed to exist.

An Analytical and Numerical Detour for the Riemann Hypothesis

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

A Review of Key Line Identification Methods in Complex Networks

This paper summarizes the key line identification method of the most representative power network in the complex networks. It is of great significance to identify the key line of the power transmission network to prevent the cascade failure of the power system. By analyzing and comparing the methods of identifying key line in recent years,this paper summarizes the power betweenness method, electrical betweenness method, comprehensive importance method,entropy theory and some other methods,and expounds their advantages and disadvantages. Based on this,the future research direction of critical line identification in complex power transmission networks is proposed.

Riemann Hypothesis and Random Walks: The Zeta Case

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 critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.