Related papers: Acid zeta function and ajoint acid zeta function
We first review our previous works of Arakawa and the authors on two, closely related single-variable zeta functions. Their special values at positive and negative integer arguments are respectively multiple zeta values and poly-Bernoulli…
In this paper, we study the relation between the zeta function of a Calabi-Yau hypersurface and the zeta function of its mirror. Two types of arithmetic relations are discovered. This motivates us to formulate two general arithmetic mirror…
By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well…
In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.
We introduce a differential topological proof and an analytical proof of Riemann hypothesis according to the saddle point method because Riemann calculated the integral representation of zeta function on the critical line by this method.…
Hypothesis of Riemann is rejected by definition, because {\zeta}(s), where s zeros of {\zeta}(s)=0, is not be equal by definition to the particular sum, which it assumes to be equal. R(s) = 1/2 holds only for the zeros of {\zeta}(s) = 0 and…
It is shown that the absolute values of Riemann's zeta function and two related functions strictly decrease when the imaginary part of the argument is fixed to any number with absolute value at least 8 and the real part of the argument is…
We study some of the interactions between the Fourier Transform and the Riemann zeta function (and Dirichlet-Dedekind-Hecke-Tate L-functions)
We consider the alternating Riemann zeta function $\zeta^*(s)= \sum^{\infty} _{ n=1} \frac{(-1)^{n-1}}{n^s}$, which converges if $Re (s)>0 .$ By using Rouche's theorem, the Bolzano-Weierstrass theorem and by method of contradiction we…
A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this…
We shall define the q-analogs of multiple zeta functions and multiple polylogarithms in this paper and study their properties, based on the work of Kaneko et al. and Schlesinger, respectively.
In this paper we have two issues coming from the same background. The first one is to describe a certain ratio of Fredholm determinants of integral operators arising from the Riemann zeta function by using the solution of a single integral…
A formal description of a functional analysis approach to the Riemann zeta-functional equation that provides in principle an infinity of different proofs based on work by the author on the existence of dilation-invariant unitary operators…
Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary…
This article provides a proof of the famous \textit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second \textit{Chebyshev Function} $\psi(x)$. We shall be extensively using complex analytic…
The extended Riemann hypothesis (ERH) for Dedekind zeta functions remains one of the most elusive open problems in number theory. Over the last century, many equivalent statements to the classical Riemann hypothesis alone have been…
This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…
A generalization of a well-known relation between the Riemann zeta function $\zeta(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli…
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…
This paper use Nevanlinna's Second Main Theorem of the value distribution theory, we got an important conclusion by Riemann hypothesis. this conclusion contradicts the Theorem 8.12 in Titchmarsh's book "Theory of the Riemann…