Related papers: The functional equation for $\zeta$
The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which…
We give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers $\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\mathbb{Z}_p$. The…
In this paper we explore the Zeta function arising from a small perturbation on a surface of revolution and the effect of this on the functional determinant and in the change of the Casimir energy associated with this configuration.
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
This note describes continued fraction representations for the rational approximations to the zeta function recently found by the author. It is tempting to think that these continued fractions might be analysed using a souped up version of…
A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.
A formula for the Hurwitz zeta function at the positive integers $k$, $\zeta(k,b)$, is created by solving the real and the imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known…
We found, by Hurwitz's Zeta Function, a new functional equation for Riemann Zeta Function. Considering this equation for $s=2$ and $s=1$, we determine a relation between the values of Riemann zeta Function on positive integers. The Matrix…
In this paper, we give a connection between the Riemann hypothesis and uniqueness of the Riemann zeta function and an analogue for L-functions.
A new definition for the Riemann zeta function for all positive integer number s > 1 is presented. We discover a most elegant expression and easy method for calculating the Riemann zeta function for small even integer values. Through this…
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta…
This paper provides some expansions of Riemann xi function, $\xi$, as a series of Bessel K functions.
We describe in detail three distinct families of generalized zeta functions built over the (nontrivial) zeros of a rather general arithmetic zeta or L-function, extending the scope of two earlier works that treated the Riemann zeros only.…
An explicit estimate for the Riemann zeta function on the critical line is derived using the van der Corput method. An explicit van der Corput lemma is presented.
A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending…
An alternative formula is presented for the evaluation of the zeta function values $\zeta(2k)$ without the need for Bernoulli numbers. Our formula is recursive, and improves the efficiency with which we can calculate large values of the…
In the present manuscript, we study analytic properties of zeta functions defined by partial Euler products.
As one of the asymptotic formulas of the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In this paper, we prove an approximate functional equation of the Barnes double zeta-function…
A proof of the Riemann hypothesis using the reflection principle is presented.
An overview of results and problems concerning the asymptotic formula for $\int_0^T|\zeta(1/2+it)|^4dt$ is given, together with a discussion of modern methods from spectral theory used in recent work on this subject.