Related papers: Another Riemann Hypothesis Equivalent
Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and M\"untz's identity.
We present a purely analytical inequality which is equivalent to the Riemann hypothesis (RH). The proof of the equivalence is based on a representation of the modulus of the Riemann $\xi$ function. As the first step to analyze the…
We use the Jacobi theta function to give a representation of the modulus of the Riemann $\xi$ function. Based on this modulus representation, we show that the Riemann hypothesis is equivalent to the validity of a family of polynomial…
This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that…
The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…
In this paper we derive some identities and inequalities on the M\"obius mu function. Our main tool is phi functions for intervals of positive integers and their unions.
Some assertions in harmonic analysis on the infinite dimensional torus are stated and their equivalence to Riemann hypothesis is proved.
In this paper, we give a connection between the Riemann hypothesis and uniqueness of the Riemann zeta function and an analogue for L-functions.
This note presents a new equivalence to the Riemann Hypothesis by means of the Salem integral equation.
We give a new equivalent condition for the Riemann hypothesis consisting in an order condition for certain finite rational combinations of the values of the Riemann zeta-function at even positive integers.
We investigate a modified M\"obius $\mu$-function which is related to an infinite product of shifted Riemann zeta-functions. We prove conditional and unconditional upper and lower bounds for its summatory function, and, finally, we discuss…
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 paper presents several new sufficient conditions, as well as new equivalent criteria for the classical Riemann Hypothesis. Noteworthy are also other statements and remarks about $\zeta$ to be found throughout the paper.
We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function.…
Using elementary means, we prove several identities involving the M\"obius function, generalizing in the multidimensional case well-known formulas coming from convolution arguments.
In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using the Li criterion.
An analog of the Riemann hypothesis is proved in this paper. Some new integral equations for the functions $\pi(x)$ and $R(x)$ follows. A new effect that is shown is that these function - with essentially different behavior - are the…
An equivalent formulation of the Riemann hypothesis is given. The physical interpretation of the Riemann hypothesis equivalent formulation is given in the framework of quantum theory terminology. One more power series related to the Riemann…
Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.
The Riemann hypothesis (RH) is well known. In this paper we would show some sufficient conditions for the RH. The first condition is related with the sum of divisors function and another one is related with the Chebyshev's function.