Regularization for zeta functions with physical applications I

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the polylogarithm function. As an immediate first application of the new parametrized series, a new expression of $\zeta$(s) follows: (s -- 1)$\zeta$(s) = -- 1 0 Li s z z -- 1 dz. As a second important application, using the functional equation and expl...

The paper was withdrawn by the author. It contained various errors.

The aim of this paper is to show further results following those published in [5], and to relate the Riemann zeta function to the relativistic cosmology.

This paper compares the distribution of zeros of the Riemann zeta function $\zeta(s)$ with those of a symmetric combination of zeta functions, denoted ${\cal T}_+(s)$, known to have all its zeros located on the critical line $\Re(s)=1/2$. Criteria are described for constructing a suitable quotient function of these, with properties advantageous for establishing an accessible proof that $\zeta(s)$ must also have all its zeros on the critical line: the celebrated Riemann hypoth...

The central idea of this article is to introduce and prove a special form of the zeta function as proof of Riemann's last theorem. The newly proposed zeta function contains two sub functions, namely $f_1(b,s)$ and $f_2(b,s)$. The unique property of $\zeta(s)=f_1(b,s)-f_2(b,s)$ is that as tends toward infinity the equality $\zeta(s)=\zeta(1-s)$ is transformed into an exponential expression for the zeros of the zeta function. At the limiting point, we simply deduce that the exp...

In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas underlying the theory. This article is going to appear in the collected works of Erich K\"ahler.

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 solutions of the similar integral equations. \noindent This paper is the English version of the paper of reference \cite{1}.

Expressing Weierstrass type infinite products in terms of Stieltjes integrals is discussed. The asymptotic behavior of particular types of infinite products is compared against the asymptotic behavior of the entire function Xi(s), well-known in Riemann zeta function theory. An approximate formula for the distribution of the non-trivial roots of Riemann's zeta function is obtained.

In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $\sigma=1$.

In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip.