# Questions tagged [dirichlet-series]

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### Is $\frac{1}{L(1+it)}$ unbounded?

Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
188 views

### Mean value theorem for Dirichlet series - optimize?

Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality $$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq \sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$ ...
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### What is the critical exponent for irregular function in the Sobolev scale?

When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and ...
136 views

### Frequency of large values of the Mertens function

It is known that with $M(x) = \sum_{n\le x}\mu(n)$, there are infinitely many $x$ s.t. $|M(x)|\ge x^{\frac{1}{2} - \varepsilon}$ (see Chapter 15 of Montgomery-Vaughan, for example). Is there any way ...
382 views

### Modern treatment of Delange's Tauberian Theorem

Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper: H. Delange - Généralisation du théorème de ...
1 vote
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### Abscissa of convergence of transformed Dirichlet series

Let $$F(s)=\sum_{k=1}^\infty \frac{f(k)}{k^s} \mbox{ and }F^*(s)=\sum_{k=2}^\infty \frac{f(k)g(k)}{k^s},$$ where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
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### Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?

The Selberg class of $L$-functions are Dirichlet series $$L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$ satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
1 vote
217 views

### Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$

I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
346 views

### Reference for behavior of Artin $L$-functions at $\Re(s) = 1$

Would anyone know a reference that proves the basic facts about Artin $L$-functions at $\Re(s) = 1$? Namely, the non-vanishing and holomorphicity for non-trivial characters. I assume this was done in ...
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### Dirichlet series of powers of the prime omega function

Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$. I was wondering what is known about the dirichlet series $$\sum_{n=1}^{\infty}\frac{\omega(n)^k}{n^s},$$ in ...
413 views

### Approximation of $\sum_{\rho}\frac{1}{|\rho|^2}$, over the non-trivial zeros of the Ramanujan's zeta function

I would like to know if it in the literature an approximation for $$\sum_{\rho}\frac{1}{|\rho|^2}\tag{1}$$ where the sum is over all of the non-trivial zeros of the Ramanujan's zeta function (also ...
1 vote