Related papers: No zeros of the partial theta function in the unit…

It is shown that there is an absolute constant $C$ such that any rational $\frac bq\in]0, 1[, (b, q)=1$, admits a representation as a finite sum $\frac bq=\sum_\alpha\frac {b_\alpha}{q_\alpha}$ where $\sum_\alpha\sum_ia_i(\frac…

In this article, we prove an explicit bound for $N(\sigma,T)$, the number of zeros of the Riemann zeta function satisfying $\sigma < \Re s <1 $ and $0 < \Im s < T$. This result provides a significant improvement over Rosser's bound for…

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain with $0\in\partial\Omega$. In this paper, we prove that for any $\beta\in(0,1)$, the supremum $$\sup_{u\in W^{1,2}(\Omega), \int_\Omega u dx=0, \int_\Omega|\nabla…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet…

Following P. Fenton, we investigate sum of translates functions $F(\mathbf{x},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to {\underline{\mathbb{R}}}:=\mathbb{R}\cup\{-\infty\}$ is a "sufficiently non-degenerate" and upper-bounded…

The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of…

Sums of the form add((-1)^n q^(n(n-1)/2) x^n, n>=0) are called partial theta functions. In his lost notebook, Ramanujan recorded many identities for those functions. In 2003, Warnaar found an elegant formula for a sum of two partial theta…

There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots-A$ indices, that is, the indices are the same. Though the way to treat $q$-multiple zeta values unless the indices are the…

In this paper, we use regularized theta liftings to construct weak Maass forms weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and Imamoglu on cycle…

We prove that the fourth powers of theta functions with even characteristics form a basis of the space of even theta functions of order four on a principally polarized Abelian variety without vanishing theta-null.

We obtain formulas for the coefficients of positive and negative powers of a partial theta function.

For $\frac12<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight $\omega$, we characterize those inner functions $\Theta$ for which $$\|\Theta'\|_{A^{p,q}_\omega}^q=\int_0^1 \left(\int_0^{2\pi} |\Theta'(re^{i\theta})|^p…

We obtain sufficient conditions for an exponential type entire function not to have zeros in the open lower half-plane. An exact inequality containing the real and imaginary parts of such functions and their derivatives restricted to the…

It is shown that, for every integer n>2, there exists \delta_{n}>0 such that the closure of the set of the real parts of the zeros of the nth partial sum of the Riemann zeta function \zeta_{n} contains to the interval [-\delta_{n},b^{n}].…

We prove that in any Sobolev space which is subcritical with respect to the Sobolev Embedding Theorem there exists a closed infinite dimensional linear subspace whose non zero elements are nowhere bounded functions. We also prove the…

Let $\theta_3(\tau)=1+2\sum_{\nu=1}^{\infty} q^{\nu^2}$ with $q=e^{i\pi \tau}$ denote the Thetanullwert of the Jacobi theta function \[\theta(z|\tau) \,=\,\sum_{\nu=-\infty}^{\infty} e^{\pi i\nu^2\tau + 2\pi i\nu z} \,.\] Moreover, let…

We say that a function $f:[0,1]\rightarrow \R$ is \emph{nowhere $L^q$} if, for each nonvoid open subset $U$ of $[0,1]$, the restriction $f|_U$ is not in $L^q(U)$. For a fixed $1 \leq p <\infty$, we will show that the set $$ S_p\doteq {f \in…

Let $f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}q^{n(n-1)/2}x^n$ ($0<q<1$) be the deformed exponential function. It is known that the zeros of $f(x)$ are real and form a negative decreasing sequence $(x_k)$ ($k\ge 1$). We investigate the complete…

This theorem is based on holomorphy of studied functions and the fact that near a singularity point the real part of some rational function can take an arbitrary preassigned value.

This work presents proofs of the main results of (math.QA/9808015), except those on q-Berezin transform to appear in a subsequent work. The notation and the results of (math.QA/9808037) and (math.QA/9808047) are used.