arXiv subjects
Publications and source records attributed to Andriy Bondarenko.
Let $γ$ denote the imaginary parts of complex zeros $ρ= β+iγ$ of $ζ(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{γ> 0}γ^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001.
The $2$kth pseudomoments of the Riemann zeta function $ζ(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $ζ(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like $(\log N)^{k^2}$, where $N$ is the length of the partial sum, but the true order of magnitude remains unknown when $k\le 1/2$. We deduce new Hardy--Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when $k\to\infty$. In the case $k<1/2$, we consider pseudomoments of $ζ^α(s)$ for $α>1$ and the question of whether the lower bound $(\log N)^{k^2α^2}$ known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali, and Radziwi{łł} and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by $\log N$ to a power larger than $k^2α^2$ when $k<1/e$ and $N$ is sufficiently large.
We combine our version of the resonance method with certain convolution formulas for $ζ(s)$ and $\log\, ζ(s)$. This leads to a new $Ω$ result for $|ζ(1/2+it)|$: The maximum of $|ζ(1/2+it)|$ on the interval $1 \le t \le T$ is at least $\exp\left((1+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. We also obtain conditional results for $S(t):=1/π$ times the argument of $ζ(1/2+it)$ and $S_1(t):=\int_0^t S(τ)dτ$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \sqrt{\log T \log\log\log T/\log\log T}$ and the maximum of $S_1(t)$ is at least $c_1 \sqrt{\log T \log\log\log T/(\log\log T)^3}$ on the interval $T^β \le t \le T$ whenever $0\le β< 1$.
We improve Montgomery's $Ω$-results for $|ζ(σ+it)|$ in the strip $1/2<σ<1$ and give in particular lower bounds for the maximum of $|ζ(σ+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $σ$. We give similar lower bounds for the maximum of $|\sum_{n\le x} n^{-1/2-it}|$ on intervals of length much larger than $x$. We rely on our recent work on lower bounds for maxima of $|ζ(1/2+it)|$ on long intervals, as well as work of Soundararajan, Gál, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.
It is shown that the maximum of $|ζ(1/2+it)|$ on the interval $T^{1/2}\le t \le T$ is at least $\exp\left((1/\sqrt{2}+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. Our proof uses Soundararajan's resonance method and a certain large GCD sum. The method of proof shows that the absolute constant $A$ in the inequality \[ \sup_{1\le n_1<\cdots < n_N} \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N \exp\left(A\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right), \] established in a recent paper of ours, cannot be taken smaller than $1$.
We prove that \[ \sum_{k,{\ell}=1}^N\frac{(n_k,n_{\ell})^{2α}}{(n_k n_{\ell})^α} \ll N^{2-2α} (\log N)^{b(α)} \] holds for arbitrary integers $1\le n_1<\cdots < n_N$ and $0<α<1/2$ and show by an example that this bound is optimal, up to the precise value of the exponent $b(α)$. This estimate complements recent results for $1/2\le α\le 1$ and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when $0<α<1/2$.
We consider the random functions $S_N(z):=\sum_{n=1}^N z(n) $, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that ${\Bbb E} |S_N|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $({\Bbb E} |S_N|^q)^{1/q}\gg_{q} \sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.
The $L^q$ norm of a Dirichlet polynomial $F(s)=\sum_{n=1}^{N} a_n n^{-s}$ is defined as \[\| F\|_q:=(\lim_{T\to\infty}\frac{1}{T}\int_{0}^T |F(it)|^qdt)^{1/q}\] for $0<q<\infty$. It is shown that \[ (\sum_{n=1}^{N} |a_n|^2|μ(n)|[d(n)]^{\frac{\log q}{\log 2} -1})^{1/2}\le \| F\|_q \] when $0<q<2$; here $μ$ is the Möbius function and $d$ the divisor function. This result is used to prove that the $L^q$ norm of $D_N(s):=\sum_{n=1}^{N} n^{-1/2-s}$ satisfies $\|D_N\|_q\gg (\log N)^{q/4}$ for $0<q<\infty$. By Helson's generalization of the M. Riesz theorem on the conjugation operator, the reverse inequality $\|D_N\|_q \ll (\log N)^{q/4}$ is shown to be valid in the range $1<q<\infty$. Similar bounds are found for a fairly large class of Dirichlet series including, on one of Selberg's conjectures, the Selberg class of $L$-functions.
It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1/\sqrt{\log\log\log N}$. The proof relies on ideas from classical work of Gál, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.
We answer Totik's question on weighted Bernstein's inequalities showing that $$ \|T_n'\|_{L_p(ω)} \le C(p,ω)\, {n}\,\|T_n\|_{L_p(ω)},\qquad 0<p\le \infty, $$ holds for all trigonometric polynomials $T_n$ and certain nondoubling weights $ω$. Moreover, we find necessary conditions on $ω$ for Bernstein's inequality to hold. We also prove weighted Bernstein-Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.
For each $N\ge C_dt^d$ we prove the existence of a well separated spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $C_d$ is a constant depending only on $d$.
In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.
We obtain an analog of Shvedov theorem for convex multivariate approximation by algebras of continuous functions.
In this short note we propose a new method for construction new nice arrangement on the sphere $S^d$ using the spaces of spherical harmonic.