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3 votes
2 answers
546 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 ...
user142929's user avatar
1 vote
0 answers
102 views

Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune

In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
user142929's user avatar
0 votes
1 answer
239 views

Zeros of partial sums of the Ramanujan's zeta function

In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia Ramanujan tau function, and we consider partial sums of its corresponding Dirichlet series (see for example the article ...
user142929's user avatar
0 votes
0 answers
199 views

List of properties of Twin primes Dirichlet series

In a paper R. Arenstorf - There are infinitely many prime twins he stated the following Dirichlet series : $$ T(s) = \sum_{n=1}^\infty \frac{\Lambda(n)\Lambda(n+2)}{n^s} $$ Question : What are ...
TPC's user avatar
  • 790
1 vote
0 answers
92 views

convergence abcissa for Mellin transforms

Where can I find the theory of abcissa of convergence for integrals necessary to understand ChenClass answer to On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$ ? Note that the ...
Patrick Sole's user avatar
0 votes
0 answers
64 views

Examples of geometrical interpretations for sequences of particular values of Dirichlet series

The remark [1] (in Spanish) shows a geometric interpretation (linking two sequences) of particular values of a given Dirichlet series, that are $\zeta(k)$ and $\zeta(2k)$. I wondered about if it is ...
user142929's user avatar
1 vote
0 answers
129 views

Asymptotic of $\sum_{1\leq n\leq x}a_n$ where $\exp(\sum_{n=1}^\infty\alpha\operatorname{rad}(n)n^{-s})=\sum_{n=1}^\infty\frac{a_n}{n^s}$

Yesterday I tried to study the article [1] in wich were showed incredible expressions related to Dirichlet series. In the same way I wondered about next question. We denote for integers $m>1$ the ...
user142929's user avatar
3 votes
2 answers
451 views

On the relation between the asymptotics of a Dirichlet series' coefficients and the series' analytic continuability

There is a wonderful series of articles by Flajolet et. al. about Mellin Transforms and the asymptotic analysis of generating functions. In particular, on page 45 of the article Mellin Transforms and ...
MCS's user avatar
  • 1,284
1 vote
1 answer
288 views

Bounding Coefficients of Dirichlet Series

Consider the exponentiated Riemann-Zeta function $\zeta(s)^p$. If it is represented as $$\zeta(s)^p = \sum_{n=1}^\infty\frac{a_n}{n^s}$$ Is there any upper bound we can put on $|a_n|$ in terms of ...
Halbort's user avatar
  • 1,129
4 votes
1 answer
448 views

The sign of an interesting sum involving a Dirichlet character

Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ). For example for $q=5$ we have \begin{equation} \begin{aligned} \chi_{5,1}&=(1, 1, 1, 1, 0),\\ ...
mike's user avatar
  • 603
4 votes
1 answer
244 views

The function $\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^s}$: reference request or particular values at integers and abscissa of convergence

We denote for integers $m\geq 1$ the Möbius function as $\mu(m)$. With the help of a CAS, Wolfram Alpha online calculator, I was calculating certain values of $$\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{...
user142929's user avatar
12 votes
3 answers
2k views

Are L-functions uniquely determined by their values at negative integers?

Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that the corresponding L-function $$L_{\{a_n\}...
Henry's user avatar
  • 1,430
16 votes
1 answer
673 views

Dirichlet series with a single zero

I need to find a Dirichlet series f that has the following property. f is zero in only one point s such that Re(s) > $\sigma_c $.
Clueless's user avatar
  • 161
7 votes
2 answers
940 views

Extracting Dirichlet series coefficients

Cauchy's integral formula is a powerful method to extract the $n$'th power series coefficient of an analytic function by evaluating a single complex integral. Is there any such analytic method to ...
MCH's user avatar
  • 1,324
3 votes
0 answers
97 views

Supremum of certain modified zeta functions at 1

Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by $$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$ where $(D/m)$ denotes the Jacobi symbol. ...
Davide Cesare Veniani's user avatar
4 votes
2 answers
566 views

Real non trivial zeros of Dirichlet L-functions

When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
A. Bailleul's user avatar
  • 1,322
8 votes
4 answers
3k views

Modern Algebraic Geometry and Analytic Number Theory

I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (...
2 votes
1 answer
178 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 ...
user119197's user avatar
4 votes
1 answer
387 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)}...
Matey Math's user avatar
2 votes
1 answer
454 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 ...
Shree's user avatar
  • 203
3 votes
0 answers
63 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^...
Craig Franze's user avatar
1 vote
1 answer
419 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 ...
Q_p's user avatar
  • 1,019
3 votes
1 answer
164 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 \[ \sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}. \]...
Mr. SnowRemover's user avatar
0 votes
1 answer
290 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 ...
Mr. SnowRemover's user avatar
1 vote
1 answer
230 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 \& ...
Alexandre's user avatar
  • 634
2 votes
0 answers
81 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 ...
David Hughes's user avatar
-3 votes
1 answer
250 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 ...
Sylvain JULIEN's user avatar
2 votes
1 answer
150 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 ...
Shivin Srivastava's user avatar
6 votes
1 answer
350 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 ...
user112214's user avatar
4 votes
0 answers
216 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 $\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, ...
Eric Towers's user avatar
8 votes
0 answers
104 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)}...
user avatar
3 votes
1 answer
560 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_{...
Adam's user avatar
  • 93
0 votes
2 answers
392 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 $...
Adam's user avatar
  • 93
6 votes
1 answer
510 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)}{...
Sigma Park's user avatar
1 vote
0 answers
325 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 $L(s,\chi)$...
davidlowryduda's user avatar
0 votes
1 answer
356 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 ...
Adam's user avatar
  • 93
2 votes
0 answers
451 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>...
user219023's user avatar
2 votes
0 answers
425 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}^\...
drewdles's user avatar
5 votes
1 answer
202 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) ...
Salvo Tringali's user avatar
3 votes
0 answers
192 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(...
davidlowryduda's user avatar
5 votes
1 answer
1k 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: $$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$ ...
Agno's user avatar
  • 4,169
1 vote
0 answers
113 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 ...
davidlowryduda's user avatar
4 votes
0 answers
318 views

Equivalence of Euler products of Dirichlet series and Meromorphic continuation

Suppose $(f_n(s))_n$ and $(g_n(s))_n$ are two sequences of Dirichlet series with positive coefficients such that $\exists \alpha\in\mathbb R$ such that for all $s\in\mathbb C$ with $\Re(s)>\alpha$ ...
kneidell's user avatar
  • 993
5 votes
0 answers
326 views

Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ : $$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) = \frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
Bertrand's user avatar
  • 1,199
12 votes
2 answers
382 views

Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.

Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that $...
George Shakan's user avatar
3 votes
1 answer
963 views

Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
edouard.gilles's user avatar
4 votes
1 answer
441 views

A lower bound on the $L^2$ norm of a Dirichlet polynomial

The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below \begin{equation*} \int_0^{T} \Big| \sum_{\alpha T &...
Matt Young's user avatar
  • 4,671
2 votes
1 answer
1k views

On the convergence of Dirichlet series over the Mobius Mu function

It is known that if $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 1/2$ then RH holds. My question is: Under RH why is it not $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = ...
Roupam Ghosh's user avatar
9 votes
2 answers
559 views

Recovering $\sum_{n \leq x} a(n)$ from $\sum_{n \leq x} a(n)e^{-n/x}$

In the theory of automorphic forms and multiple Dirichlet series, we often take inverse Mellin transforms of Dirichlet series to come up with Tauberian theorems, like the Ikehara Tauberian method. In ...
davidlowryduda's user avatar
14 votes
0 answers
1k views

Analytic continuation of the Dirichlet generating series of the multiplicative partition function

Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series: $$\kappa(s)=\prod_{m=2}^{\infty}\frac{1}{1-m^{-s}}=\sum_{n=1}^{\infty}\frac{\...
mohammad-83's user avatar