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Abstract

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.