# Questions tagged [wieners-tauberian-theorem]

Intended to label questions regarding Wiener's Tauberian Theorem and related topics.

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### Are these two definitions of the Mellin transform equivalent?

I've been reading Chapters 6 and 7 of Bateman and Diamond's Analytic Number Theory for a more formal treatment of Mellin transforms than I've seen previously. However, I'm puzzled by the fact that, ...
• 480
776 views

### Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit $$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$ It is clear that the ...
• 915
1 vote
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### Is there a dilation/scaling "equivalent" of Wiener's Tauberian theorems?

Given an even function $g\in L^2(\mathbb{R})$, if it is orthogonal to $\exp(-\alpha^2 x^2)$ for all $\alpha$ (really, this is only needed on an open neighborhood of $1$), then it is identically zero. ...
• 29.8k
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For $a>0$ let us define $$H^2(-a,a)=\{f \ \mbox{is analytic in the strip |\Im(z)|<a}: \sup_{y\in [-a,a]}\int_{\mathbb{R}}|f(x+iy)|^2\,dx<\infty\}.$$ For $f\in H^2(-a,a)$, define $\|f\|=\... • 742 2 votes 1 answer 144 views ### How to generalize Newman's simplification of O-Tauberian theorem? Can you prove the next theorem: Let$f$be Dirichlet series with real, positive coefficients$(a_n>0)$. If$f$is holomorphic on$\Re(z)\ge1$, but has one singularity at$z=1$, then$\lim_\limits{...
• 2,190
1 vote
I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form $$\sum_{n\leq X} a(n) = cX^a(\log X)^m + O\big(X(\log X)^{m-1}\big),$$ i.e. ...