Related papers: The functional equation for $\zeta$
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…
Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential…
In this paper we introduce new generalizations of the zeta function, the Tricomi functions; their main properties are studied. This opens the way to a deeper, better application of these functions both in the theory of special functions,…
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…
We express the Riemann zeta function $\zeta\left(s\right)$ of argument $s=\sigma+i\tau$ with imaginary part $\tau$ in terms of three absolutely convergent series. The resulting simple algorithm allows to compute, to arbitrary precision,…
We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature.…
For a scheme $X$ whose $\mathbb F_q$-rational points are counted by a polynomial $N(q)=\sum a_iq^i$, the $\mathbb{F}_1$-zeta function is defined as $\zeta(s)=\prod(s-i)^{-a_i}$. Define $\chi=N(1)$. In this paper we show that if $X$ is a…
We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the lambda-structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the…
We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.
It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.
We obtain formulas for the coefficients of positive and negative powers of a partial theta function.
New formulas for approximation of zeta-constants were derived on the basis of a number-theoretic approach constructed for the irrationality proof of certain classical constants. Using these formulas it's possible to approximate certain…
A functional equation between the zeta distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.
We will prove that the zeta function for Ruelle-expanding maps is rational.
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.
Commenting on an observation of Prof. Edwards, this note presents a method of evaluation of $\zeta(2n)$ that follows easily from Riemann's own representation of the zeta function.
We define reduced zeta functions of Lie algebras, which can be derived from motivic zeta functions using the Euler characteristic. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to…
This short note contributes a new zero-free region of the zeta function. This zero-free region has the form {s : Re(s) > a}, where a = 21/40.
In this paper, we construct the alternating multiple q-zeta function(= Multiple Euler q-zeta function) and investigate their properties. Finally, we give some interesting functional eauations related to q-Euler polynomials.
In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann's Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work "Remarques…