相关论文: Rank Two Non-Abelian Zeta and Its Zeros

J. R. Wilton obtained an expression for the product of two Riemann zeta functions. This expression played a crucial role to find the approximate functional equation for the product of two Riemann zeta functions in the critical region. We…

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…

Let $\alpha>1$ be an irrational number of finite type $\tau$. In this paper, we introduce and study a zeta function $Z_\alpha^\sharp(r,q;s)$ that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the…

We show that the analytic continuations of Helson zeta functions $ \zeta_\chi (s)= \sum_1^{\infty}\chi(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole…

The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its…

We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large…

Let $\zeta(s)$ and $Z(t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_0^T Z(t)\zeta(1/2+it)dt$ and $\int_0^T Z^2(t) \zeta(1/2+it)dt$. Furthermore we derive an upper bound for…

In this manuscript, we show that the Riemann zeta function satisfies $\big(\zeta(s),\zeta(1-\overline{s})\big)\neq(0,0)$ for any $s$ in the critical strip, except on the critical line. This still holds even when the fractional part function…

Two representations of the Bessel zeta function are investigated. An incomplete representation is constructed using contour integration and an integral representation due to Hawkins is fully evaluated (analytically continued) to produce two…

Results on the upper and lower semicontinuity of functionals defined on spaces of convex and more general functions are established. In particular, the following result is obtained. Let $\phi(v; \cdot)$ be the density of the absolutely…

The Argand diagram is used to display some characteristics of the Riemann Zeta function. The zeros of the Zeta function on the complex plane give rise to an infinite sequence of closed loops, all passing through the origin of the diagram.…

In this note we investigate the existence of zeros of linear twists of $L$-functions outside of the critical strip. In particular, we show that the Lerch zeta function $L(\lambda,\alpha,s)$ has infinitely many zeros for $1<\sigma<1+\eta$,…

This paper shows that, in the critical strip, the Riemann zeta function $\zeta(s)$ have the same set of zeros as $F(s):=\int_0^\infty t^{s-1}(e^t+1)^{-1}dt$, and then discusses the behavior of $F(s)$.

In 2007 Chang and Yu determined all the algebraic relations among Goss's zeta values for the rational function field - these are also known as the Carlitz zeta values. Goss raised the problem about algebraic relations among Goss's zeta…

We consider the variation of two fundamental types of zeta functions that arise in the study of both physical and analytical problems in geometric settings involving conical singularities. These are the Barnes zeta functions and the Bessel…

This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…

We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adele class space of the field of…

We give a conditional lower bound on the number of non-trivial simple zeros for the Dedekind zeta function $\zeta_{K}(s)$, where $K$ is a quadratic number field. The conditional result is given by assuming a Lindel\"of on average (in the…

We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any…

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…