Related papers: On the complex conjugate zeros of the partial thet…

Ramanujan presented four identities for third order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. provided different proofs by…

In the present paper we give a simple mathematical foundation for describing the zeros of the Selberg zeta functions $Z_X$ for certain very symmetric infinite area surfaces $X$. For definiteness, we consider the case of three funneled…

It is shown that Weng's zeta functions associated with arbitrary semisimple algebraic groups defined over the rational number field and their maximal parabolic subgroups satisfy the functional equations.

We prove a formula for the Mangoldt function which relates it to a sum over all the non-trivial zeros of the Riemann zeta function, in addition we analize a truncated version of it.

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|,…

The Riemann $\Xi(z)$ function (even in $z$) admits a Fourier transform of an even kernel $\Phi(t)=4e^{9t/2}\theta''(e^{2t})+6e^{5t/2}\theta'(e^{2t})$. Here $\theta(x):=\theta_3(0,ix)$ and $\theta_3(0,z)$ is a Jacobi theta function, a…

We investigate the proportion of the nontrivial roots of the equation $\zeta (s)=a$, which lie on the line $\Re s=1/2$ for $a \in \mathbb C$ not equal to zero. We show that at most one-half of these points lie on the line $\Re s=1/2$.…

We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two…

We show that the gradient and the hessian of the Riemann theta function in dimension n can be combined to give a theta function of order n+1 and modular weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus of this…

This paper deals with the study of the zeros of the big $q$-Bessel functions. In particular, we prove a new orthogonality relations for this functions similar to the one for the classical Bessel functions. Also we give some applications…

In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent…

Let $\mathbf{G}$ be a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. We consider bivariate zeta functions of groups of the form $\mathbf{G}(\mathcal{O})$…

We introduce the multiple zeta functions with structures similar to those of symmetric functions such as Schur $P$-, Schur $Q$-, symplectic and orthogonal functions in the representation theory. We first consider their basic properties such…

We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of…

Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1…

We find zero-free regions in the complex plane at large |q| for the multivariate Tutte polynomial (also known in statistical mechanics as the Potts-model partition function) Z_G(q,w) of a graph G with general complex edge weights w = {w_e}.…

This note studies the Laurent series of the inverse zeta function $1/\zeta(s)$ at any fixed nontrivial zero $\rho$ of the zeta function $\zeta(s)$, and its connection to the simplicity of the nontrivial zeros.

We consider the set W of double zeros in (0,1) for power series with coefficients in {-1,0,1}. We prove that W is disconnected, and estimate the minimum of W with high accuracy. We also show that [2^(-1/2)-e,1) is contained in W for some…

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell--Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving…

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| >…