Related papers: Another Riemann Hypothesis Equivalent
Basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta- functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of…
New unconditional estimates of the divisor and totient functions are contributed to the literature. These results are consistent with the Riemann hypothesis and seem to solve the Nicolas inequality for all sufficiently large integers.
New simple proofs are given to some elementary approximate and explicit inversion formulas for Riesz potentials. The results are applied to reconstruction of functions from their integrals over Euclidean planes in integral geometry.
We study the existence of Riemann-Stieltjes integrals of bounded functions against a given integrator. We are also concerned with the possibility of computing the resulting integrals by means of related Riemann integrals. In particular, we…
We make plausible the existence of counterexamples to the Riemann hypothesis located in the neighbourhood of unusually large peaks of $\vert \zeta \vert$. The main ingredient in our argument is an identity which links the zeros of a…
We give two distinct infinite-Hamiltonian representations for the Riemann equation. One with first order Hamiltonian operators and another with third order-first order Hamiltonian operators. Both representations contain an arbitrary…
Similarity metric which is not positive definite, and present a general theorem which provides a large family of similarity metrics which are positive definite.
We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.
The paper provides the proof of the Rimann's conjecture. The results of the works of A. M. Odlyzko and H. te Riile "Disproof of the Conjecture", which gives a disproof of the Mertens hypothesis, using to prove the Riemann's hypothesis. This…
Let the summatory function of the M\"{o}bius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the…
We find conditions equivalent to some commutator identities considered in Part I
Physics is a fertile environment for trying to solve some number theory problems. In particular, several tentative of linking the zeros of the Riemann-zeta function with physical phenomena were reported. In this work, the Riemann operator…
We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm1$ and such that $M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go…
A GGC (Generalized Gamma Convolution) representation of Riemann's Xi-function is constructed.
By considering even functions (mod $n$) we generalize a Menon-type identity by Li and Kim involving additive characters of the group ${\Bbb Z}_n$. We use a different approach, based on certain convolutional identities. Some other…
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
The variable change w=exp(u) is applied to establish novel integral representations of the incomplete gamma-function, hypergeometric F-function,confluent hypergeometric /Phi-function and beta-function, and to analyze these functionsas as…
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…
We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in $R$-linear abelian categories of…