Related papers: Oscillation of irrational measure function in the …

We study the finite theta correction to the metric of the moduli space of noncommutative multi-solitons in scalar field theory in (2+1) dimensions. By solving the equation of motion up to order O(theta^{-2}) explicitly, we show that the…

We collect some peculiarities of higher-order fractional Laplacians $(-\Delta)^s$, $s>1$, with special attention to the range $s\in(1,2)$, which show their oscillatory nature. These include the failure of the polarization and…

Karamata's integral representation for slowly varying functions is extended to a broader class of the so-called $\psi$-locally constant functions, i.e. functions $f(x)>0$ having the property that, for a given non-decreasing function $\psi…

In this work we prove the existence of infinitely many nonradial solutions that change signal to the problem $-\Delta u=f(u)$ in $B$ with $u=0$ on $\partial B$, where $B$ is the unit ball in $\mathbb{R}^2$ and $f$ is a continuous and odd…

This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $\theta _1$, $\theta _2$, $\theta _3$, $\theta _4$ of one complex variable, presented in a ready-to-use form. An attempt is…

We consider difference equations with several non-monotone deviating arguments and nonnegative coefficients. The deviations (delays and advances) are, generally, unbounded. Sufficient oscillation conditions are obtained in an explicit…

In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential…

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\mathrm{o}(T)$, $$ \int_{-(\log T)^{\theta}}^{(\log T)^{\theta}} |\zeta(\tfrac 12 + \mathrm{i} t + \mathrm{i} h)|^{\beta} \mathrm{d} h = (\log…

The series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $|q|<1$ and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. It is known that for $q$ taking one of the…

Precise formulas are derived for the expected values $<\xi>$, $<\eta>$ and variances $\delta \xi ^2$, $\delta \eta ^2$ of random variables $\xi$, $\eta$ describing the spin asymmetry of some reaction when a background process contribution…

We continue the analysis of a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in\mathbb{R}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly…

We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm…

The function $\sum_{p\leq x} \frac{1}{p} - \log\log(x) - M$ is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before $\exp(495.702833165)$.

It is shown that the Jacobi and Riemann identities of degree four for the multidimensional theta functions as well as the Weierstrass identities emerge as algebraic consequences of the fundamental multidimensional binary identities…

Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ \theta (\alpha;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-\alpha \frac{\pi }{y }|mz+n|^2}$ be the theta function associated with the lattice $L ={\mathbb Z}\oplus z{\mathbb Z}$. In…

Let $a(1) >0$, $a(n) \ge 0$ for $n \ge 2$ and $a(n) = O(n^\varepsilon)$ for any $\varepsilon >0$, and put $Z(\sigma + it):= \sum_{n=1}^\infty a(n) n^{-\sigma - it}$ where $\sigma , t \in {\mathbb{R}}$. In the present paper, we show that any…

For a fixed $\theta\neq 0$, we define the twisted divisor function $$ \tau(n, \theta):=\sum_{d\mid n}d^{i\theta}\ .$$ In this article we consider the error term $\Delta(x)$ in the following asymptotic formula $$ \sum_{n\leq x}^*|\tau(n,…

We find an asymptotic expansion of a multi-dimensional version of Selberg's central limit theorem for $L$-functions on $ \sigma= \frac12 + ( \log T)^{-\theta}$ and $ t \in [ T, 2T]$, where $ 0 < \theta < \frac12 $ is a constant.

Riemann's non-differentiable function and Gauss's quadratic reciprocity law have attracted the attention of many researchers. In \cite{RM} Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta-transformation…

It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant…