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Questions tagged [dirichlet-series]

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0
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1answer
61 views

meromorphic extension of dirichlet series

Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...
4
votes
1answer
156 views

Zeros of derivatives of Dirichlet Eta function

Let $$ \eta^{(d)}(z) = \sum_{n=1}^\infty \dfrac {(-1)^d(-1)^{n-1}\ln(n)^d} {n^z} $$ be the derivative of Dirichlet Eta function of order $d$. Does it exist any known or not known zero of $\eta^{(d)}...
4
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0answers
168 views

Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\ \sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D} \text{ for all }f(z)=\sum\...
2
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1answer
252 views

Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?

Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
3
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0answers
109 views

Dirichlet eta function and Stirling Permutations

The Stirling permutations of order $k$ is a permutation of the multiset $1, 1, 2, 2, ..., k, k$. The Dirichlet $\eta$-function is a function closely related to the Riemann $\zeta$-function. According ...
3
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0answers
40 views

Analytic continuation of a Dirichlet series with several complex variables

For $w_1,w_2,z_1,z_2\in\mathbb{C}$ with $\operatorname{Re}(w_1)>0$ and $\operatorname{Re}(w_2)>0$, define \begin{equation*} U(w_1,w_2;z_1,z_2):=\prod_{p}\left(1-\frac{e^{z_1}}{p^{1+w_1}}-\frac{e^...
1
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1answer
210 views

On a certain integral representation for Dirichlet L-functions

It is an ancient result of Jensen that $$(s-1)\zeta(s)=\frac{\pi}{2} \int_{-\infty}^{\infty} \frac{(1/2+it)^{1-s}}{\cosh^{2}\pi t} \mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. Is ...
1
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0answers
53 views

Some theoretical question on Euler product

It is very rare that the Euler product \[ \lim_{X \to \infty}\prod_{p \leq X}(1 + a(p)p^{-s}) \] conditionally converges for $\sigma > A$ with some $0 < A \leq 1$ when $|a(n)| = 1$. Suppose ...
3
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1answer
147 views

Recovering information for $\sum_{n \leq x}a(n)$ from $\sum_{n \geq 1}a(n)e^{-nx}$

I am wondering if I could deduce the bound for the partial sums \[ \sum_{n \leq x}a(n) \ll x^{A}, \quad x \to \infty \] from the relation \[ \sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}. \]...
1
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0answers
59 views

Dirichlet series with an abscissa of absolute convergence $\sigma_{0}$, analytic in $\sigma > \sigma_{0} - \delta$

Suppose that a Dirichlet series $f(s)$ has the abscissa of absolute convergence $\sigma_{0}$ and is analytic in $\sigma > \sigma_{0} - \delta$ for some $\delta > 0$. For $\sigma > \sigma_{0}$,...
0
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1answer
118 views

Analytic continuation of Euler product $\prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1}$

Is anything useful known about the function defined by \[ f(s, \alpha) = \prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1} \quad ? \] Here, $\alpha$ is real. When $\alpha = 1$, this is certainly the ...
1
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1answer
149 views

Asymptotic for a number theoretic sequence and its Dirichlet series' convergence

I would like to know the asymptotic behaviour at large $n$ for $t\in\mathbb{R}$, $t\neq0$ of the following function: \begin{align*} A_n(t)&=\sum_{q=\frac{a}{b}\in \mathbb{Q}^+|\gcd(a,b)=1 \& ...
1
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0answers
39 views

Which complex maps with branch cuts have a representation by Dirichlet series?

Which complex maps with branch cuts have a representation by Dirichlet series? I am aware of the work of A.F. Leont'ev on general Dirichlet series, and the theorems of representation of analytic ...
2
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0answers
63 views

Question on a generalized Dirichlet series

Given the generalized Dirichlet series $$S(x) =\sum_{(n,m)\in \mathbb{Z}^2}e^{-x\sqrt{n^2+m^2}} $$ is there any way to solve the equation $$2S(2x)=S(x)$$ for $x\in\mathbb{R}$? I am only interested in ...
-3
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1answer
167 views

Twin prime based Dirichlet series

Assuming there are infinitely many twin primes, one can consider a Dirichlet series $ \sum_{n>0}a_{n}{n^{-s}} $ and replace the sequence of positive integers with the sequence of twin primes. That ...
2
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1answer
106 views

Residue of the following variant of Dirichlet function [closed]

I am working with the Piltz divisor problem where the number of ways in which a number $n$ can be written as a product of $k$ is of the form $$D_{k}(x)=xP_{k}(\log x)+\Delta _{k}(x)$$ where $P_k$ is ...
6
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1answer
148 views

Counting smooth numbers in short intervals

I am reading a few papers about counting smooth numbers in the interval $[x, x+\sqrt{x}]$, including the work of Harman, and Matomaki. Both authors mentioned that the Dirichlet polynomial techniques ...
0
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1answer
103 views

Calculating a Dirichlet Character

How would I calculate a function such as $$\sum_{n \leq x}r(n) = L(1, \chi) \cdot x + O(x^{1-\eta}),$$ where $r(n) = \sum_{d|n} \chi(d)$? The part I'm having difficulty calculating is the L-function ...
1
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0answers
102 views

Analytic continuation of $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s}$. Possibly related to Riemann Zeta function $\zeta(s)$?

I'm trying to find the analytic continuation for $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s} ,$ with $a\in \mathbb{N^+}$ and $s<1$. I need most likely only the values for $s=\frac{1}{2}-m$...
3
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0answers
95 views

Gap Between Abscissae of Conditional Convergence and Holomorphicity for Dirichlet Series

For a Dirichlet series, $D = \sum_n a_n n^{-s}$ we may define the abscissae, in (non-strictly) increasing order $\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, ...
7
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0answers
98 views

What is known about the following series?

For $k\in{\mathbb Z}^2$ write $|k|=\sqrt{k_1^2+k_2^2}$ for the euclidean norm. Then let $g(k)=gcd(k_1,k_2)$. For $s\in\mathbb C$ let $$ D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
2
votes
1answer
215 views

critical line inequality concerning the square of the modulus of a Dirichlet polynomial

I am currently studying the following inequality involving the square of the modulus of a specific Dirichlet polynomial: $$\left( \sum_{1}^{N}\frac{1}{n} \right)^2 \ \ - \ \left| \sum_{1}^{N}\frac{(...
14
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3answers
616 views

Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $

I am trying to prove or disprove $$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$ where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
2
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0answers
68 views

Bound for partial sums of $ L(1/2+it,\chi)$

Let $\chi$ be a primitive Dirichlet character of conductor $q>1$. One may use partial summation to prove an upper bound of the form (I hope I am right) $$ \sum_{n\leq X} \chi(n)n^{-1/2-it} \ll \...
0
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1answer
81 views

Upper bound for tail in Dirichlet series

We know the elementary fact that if the partial sums $ \sum_{n\leq X} a_n $ are bounded, say by $ C$, then the series $ \sum_{n\geq 1} a_n n^{-s} $ converges for $s >0$. My question then is, is ...
3
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1answer
195 views

The abscissa of convergence of the real part of a Dirichlet series

Let $L(s)=\sum_{n\ge1}\frac{a(n)}{n^s}$ be a Dirichlet series with a finite abscissa of convergence $\sigma_c.$ My question is the following : On what condition the abscissa of convergence of $\sum_{...
2
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2answers
204 views

Does there exist a known Dirichlet series verifying all these conditions and have non trivial zeros off the critical line

Let $s=α+iβ$ be a complex number. Consider the Dirichlet series of the form $$f(s)=∑_{n=1}^{∞}(a_{n})/n^{s}$$ where $(a_{n})_{n≥1}$ is a real sequence. We consider the class of Dirichlet series ...
0
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2answers
246 views

What is the relationship between the abscissa of holomorphy and abscissa of convergence of a Dirichlet series

Given a Dirichlet series $$\phi(s)=\sum_{n\ge1}\frac{a_n}{n^s}$$ let $\sigma_{\text{conv}}\in\bar{\mathbb{R}}$ its abscissa of convergence, then we know that $\phi(s)$ is holomorphic on the half-plan $...
5
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1answer
257 views

If a Dirichlet series converges Conditionally, how can I apply Euler product?

In 1737, Euler discovered that if $ f(n) $ is multiplicative and $ \sum f(n)/n^{s} $ converges absolutely for ${\rm Re}(s) > \sigma_a$ then we have \begin{equation} \sum_{n=1}^{\infty} \frac{f(n)}{...
2
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0answers
90 views

Harmonic Dirichlet series

The harmonic numbers are defined by $$ H_n=\sum_{j=1}^n\frac{1}{j} $$ I have come across the following sum: $$ g(z)=\sum_{n=1}^\infty z^{H_n} $$ Clearly it converges only for $z<1/e$. Is it a ...
5
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0answers
90 views

On a particular case of Dirichlet series [closed]

I've this series: $$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$ where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $. I need the limit of series like an analytic function of $...
1
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0answers
226 views

Extension of Heath-Brown's fourth moment $\sum_ \chi \lvert L (s, \chi) \rvert^4$ to complex characters

Let $L(s, \chi)$ denote the Dirichlet $L$-function associated to the character $\chi$. In his paper A Mean Value Estimate for Real Character Sums, Heath-Brown proves a mean value bound for $L(s,\chi)$...
0
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1answer
275 views

Meromorphic continuation of a Dirichlet series

I asked this question in SEM but I got no answer, so I'm trying my luck here. Let the Dirichlet series $\phi(s)=\sum_{n\ge 1}\frac{a(n)}{n^s}$ be absolutely convergent for $\Re(s)>1$ and extend to ...
2
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0answers
246 views

Analytic continuation of “composite” zeta function

Let us define the Dirichlet series $$\mathcal C(s):=\sum_{n\text{ composite}}\frac{1}{n^s},\quad P(s):=\sum_{p\text{ prime}}\frac{1}{p^s}.$$ They are absolutely convergent in the half-plane $\sigma>...
2
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0answers
418 views

Analytically continuing the limit of this series?

Main Question I believe the following formula gives the right answer: $$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times \sum_{r=1}^\...
2
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0answers
77 views

Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: https://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the ...
4
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0answers
86 views

References for “quadratic” Dirichlet series

(Please pardon the use of nonstandard terminology, as I know not the accepted names for my entities of interest.) Some personal research I have been doing led me to consider series of the form $$\...
4
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1answer
128 views

Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...
3
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0answers
109 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ \...
4
votes
1answer
269 views

How do I evaluate this sum for $s$ is a complex variable :$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$?

This question related to this question in SE ,I would like to know how do I evaluate this sum for $s$ is a complex variable :$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2s}n!}$$ . Edit01:And I think ...
1
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0answers
124 views

An apparent closed form for a slightly tweaked Dirichlet L-function. Could it be proven? [closed]

I made a small tweak to the well-known Dirichlet L-function ($p$=prime): $$L(s, \chi_4) :=\prod_p \bigg(\frac {p^s}{p^s-\chi_4(p)} \bigg)=\prod_p \bigg(\frac {p^s}{p^s-\sin\left(\frac{p \,\pi}{2}\...
2
votes
2answers
398 views

Is there a nice generating function proof of the following identity?

Consider the Jordan function $J_2(n)$ defined by $$ J_2(n) = \#\{x \in (\mathbb{Z}/n)^2 \mid ord(x) = n\} $$ (this is OEIS A007434). One can prove the following identity pretty easily: $$ \sum_{d \mid ...
2
votes
0answers
153 views

Determining coefficients of a Dirichlet series based on values on a vertical line

Let us suppose we have a Dirichlet series $$ D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$ and that we know the values of $D(\tfrac{1}{2} + im)$ for $m \in \mathbb{Z}$. Can we recover the coefficients $a(...
2
votes
1answer
212 views

Does $\prod_{n=2}^{\infty} \left(\frac {1}{1-\frac{\chi_k(n)}{n^s}} \right)$ converge for non-principal characters for all $\Re(s) > \frac12$?

This question loosely builds on this one. Take the following infinite product: $$N(s,\chi_k)=\prod_{n=2}^{\infty} \left(\frac {1}{1-\dfrac{\chi_k(n)}{n^s}} \right)$$ with $\chi_k$ a Dirichlet ...
4
votes
1answer
475 views

Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE. In the literature about Dirichlet $L$-series, I found that their Euler products: $$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$ ...
1
vote
0answers
240 views

Smoothing Dirichlet Series partial sums by means of Pontifex Path Bending Functions

I have recently stumbled upon a curious approach to smoothing of the partial sums of Ordinary Dirichlet Series, and which I would like to share with others. Theorem 11.18 of Apostol's "Introduction to ...
2
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0answers
72 views

Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance: $y[n] = x[n] + y[n-1]$ $Y(z) = X(z) + Y(z) \...
1
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0answers
106 views

Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?) But I would like to know if ...
1
vote
1answer
161 views

Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like : $$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...
1
vote
0answers
100 views

Extracting information from $\sum_{n \leq X} a(n) (X-n)^d$

Allow me to give context to the question, which appears in the box at the bottom. A very general hope from the theory of Dirichlet series is to try to extract information about the coefficients of a ...