Related papers: Generalized Bombieri - Lagarias' theorem and gener…

Recently, we have established the generalized Li's criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_rho(1-(1-((rho-a)/(rho+a-1))^n) for any real a not…

Recently, we have established the generalized Li's criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,b=Sum_rho(1-(1-((rho+b)/(rho-b-1))**n) for any real b not…

Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not…

Recently, in arXiv:1304.7895; Ukrainian Math. J. - 2014.- 66. - P. 371 - 383, we presented the generalized Li's criterion. This is the statement that the sums /lambda_(n, b, /sigma)=Sum_(rho)((1-((/rho+b)/(/rho-b-2*/sigma))^n)), taken over…

We present the first applications of the recently established by us (arXiv:1304.7895; Ukrainian Math. J. - 2014. -66. - P. 371-383) generalized Li's criterion equivalent to the Riemann Hypothesis. This criterion is the statement that the…

Recently, Voros has found the sums involving certain powers of z-1/2, which, when taken over Riemann xi-function zeroes /rho, must be positive for the Riemann hypothesis holds true and vice versa. Here we analyze these sums, write them as…

Let $F$ be a function in the Selberg class ${\mathcal S}$ and $a$ be a real number not equal to 1/2. Consider the sum $$\lambda_{F}(n,a)=\sum_{\rho}\left[1-\left(\frac{\rho-a}{\rho+a-1}\right)^{n}\right],$$ where $\rho$ runs over the…

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants c_n, n =…

For each positive integer $n$, function $f$, and point $x$, the 1998 conjecture by Ghinchev, Guerragio, and Rocca states that the existence of the $n$-th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of all $n(n+1)/2$…

Robin's criterion states that the Riemann hypothesis is equivalent to $\sigma(n) < e^\gamma n \log\log n$ for all integers $n \geq 5041$, where $\sigma(n)$ is the sum of divisors of $n$ and $\gamma$ is the Euler-Mascheroni constant. We…

In this paper, we make use of Robin and Lagarias' criteria to prove Riemann hypothesis. The goal is, using Lagarias criterion for $n\geq 1$ since Lagarias criterion states that Riemann hypothesis holds if and only if the inequality…

Li showed that the Riemann Hypothesis is equivalent to the nonnegativity of a certain sequence of numbers. Bombieri and Lagarias gave an arithmetic formula for the number sequence based on the Guinand-Weil explicit formula and showed that…

We extend the notion of generalised Cesaro summation/convergence developed previously to the more natural setting of what we call "remainder" Cesaro summation/convergence and, after illustrating the utility of this approach in deriving…

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed $\alpha \in \Q - \Z_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre…

Exact and asymptotic formulae are displayed for the coefficients $\lambda_n$ used in Li's criterion for the Riemann Hypothesis. In particular, we argue that if (and only if) the Hypothesis is true, $\lambda_n \sim n(A \log n +B)$ for $n \to…

In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li…

In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet $L$-function $L(s,\chi)$ is equivalent to the following bound: Let $k \geq 1$ and $\ell$ be positive real numbers. For any $\epsilon >0$, we have…

The Riemann Hypothesis states that within the strip region of the complex plane $0 < {\rm Re}(s) < 1$, the Riemann $\xi(s)$ function has zeros only on the critical line ${\rm Re}(s) = \frac{1}{2}$ and none elsewhere. To prove the Riemann…

In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function for $1/2<\sigma<1$ and sufficiently large $t$. We give an explicit upper and lower bounds for $\Re\sum_{\rho}…