Exact Wiener–Ikehara theorems

Wen-Bin Zhang¹

¹ Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A. and Department of Mathematics South China University of Technology Guangzhou, People’s Republic of China

Acta Arithmetica, Tome 187 (2019) no. 4, pp. 357-380

Résumé

A function $f(x)$ defined on $[a,\infty)$, $a\ge 0$, is said to be linearly slowly decreasing with index $\alpha \ge 0$ if \[ \liminf_{x\to \infty,\, y/x\to 1+} (x\log^\alpha x)^{-1}(f(y)-f(x))\ge 0. \] Nondecreasing functions and slowly decreasing functions are linearly slowly decreasing with index $\alpha=0$; but the converse is not true. Let $F(x)$ be a real-valued Lebesgue measurable function with support in $[0,\infty)$ such that \[ \int_0^\infty e^{-\sigma x}|F(x)|\,dx \lt \infty, \quad\ \sigma \gt 1. \] Let $\varDelta_\lambda^{*m}(t)$ be the $m$-fold convolution of \[ \varDelta_\lambda(t):=(1-|t|/(2\lambda))^+/2, \quad t\in\mathbb{R}, \] with itself. We prove the following exact Wiener–Ikehara theorem. $\mathbf{Theorem.}$ $\lim_{x\to \infty}(e^x x^\alpha)^{-1}F(x)=L/\varGamma(\alpha+1)$ if $F(\log\,u)$ is a linearly slowly decreasing function of $u$ with index $\alpha$ and there exist a constant $\lambda_0\ge 0$ and a positive integer $m\ge 1+ [\alpha]/2$ such that, for every $\lambda \gt \lambda_0$, \[ \frac{1}{y^{\alpha}}\int_{-\infty}^{\infty}\varDelta_\lambda^{*m}(t)e^{ity}(G(\sigma+it)-G(\sigma^\prime+it))\,dt \] approaches zero as $\sigma,\,\sigma^\prime \to 1+$ uniformly for $y\ge y_0(\lambda)\, ( \gt 0)$, where \[ G(s)=\int_0^\infty e^{-sx}F(x)\,dx-\frac{L}{(s-1)^{\alpha+1}}. \] Conversely, if $\lim_{x\to\infty}(e^x x^\alpha)^{-1}F(x)=L/\varGamma(\alpha+1)$ then all the conditions hold for all $\lambda \gt 0$ and every integer $m\ge 1+[\alpha]/2$. The classical Wiener–Ikehara theorem is a special case of the above theorem with nondecreasing function $F(x)$. We also prove a Wiener–Ikehara upper bound theorem and a Wiener–Ikehara lower bound theorem. Preliminary applications to Beurling generalized primes are briefly discussed too.

Détail

DOI : 10.4064/aa171208-7-6

Keywords: function defined infty said linearly slowly decreasing index alpha liminf infty log alpha f nondecreasing functions slowly decreasing functions linearly slowly decreasing index alpha converse real valued lebesgue measurable function support infty int infty sigma infty quad sigma vardelta lambda *m m fold convolution vardelta lambda lambda quad mathbb itself prove following exact wiener ikehara theorem mathbf theorem lim infty alpha vargamma alpha log linearly slowly decreasing function index alpha there exist constant lambda positive integer alpha every lambda lambda frac alpha int infty infty vardelta lambda *m ity sigma g sigma prime approaches zero sigma sigma prime uniformly lambda where int infty sx dx frac s alpha conversely lim infty alpha vargamma alpha conditions lambda every integer alpha classical wiener ikehara theorem special above theorem nondecreasing function prove wiener ikehara upper bound theorem wiener ikehara lower bound theorem preliminary applications beurling generalized primes briefly discussed too