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14 votes
0 answers
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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
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
6 votes
0 answers
286 views

Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum

Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
H A Helfgott's user avatar
  • 20.2k
6 votes
0 answers
233 views

Mean value theorem for Dirichlet series of prime support?

Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973), $$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$ ...
H A Helfgott's user avatar
  • 20.2k
6 votes
0 answers
654 views

Generalized prime number theorem and Riemann Hypothesis for non-number math objects

My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
Vincent Granville's user avatar
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
4 votes
0 answers
134 views

Converse theorem for zeta universality

Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
modperspec'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
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
3 votes
0 answers
167 views

A sharper estimate for a generalization of the sum-of-divisors function

I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$ This ...
Bear's user avatar
  • 31
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
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
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
2 votes
0 answers
131 views

Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?

I am trying to get an asymptotic formula such as $$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$ where $L_4(s, n)$ is the first $n$...
Vincent Granville's user avatar
2 votes
0 answers
158 views

Error or gap in "Modular Functions and Dirichlet Series", by Apostol

My question concerns Apostol's Chapter 7, Kronecker's Theorem with Applications. It's Theorem 7.11, page 156. I’m attaching the proof in question. There is a lot going on, but I’ve highlighted the ...
Lawrence Paulson's user avatar
2 votes
0 answers
188 views

How to best approximate $1/\zeta(s)$ by a finite sum

I would like to approximate $1/\zeta(s)$ for $s=1+it$ by a finite sum: $$\frac{1}{\zeta(1+it)} = \sum_n \frac{\mu(n)}{n} \eta\left(\frac{n}{x}\right) + \epsilon(t)$$ with $\eta$ a function of compact ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
194 views

On a generalization of the Möbius function from number theory

Let $\omega$ be a positive real number, and define: $$\mathbf{1}_{\omega}\left(n\right)\overset{\textrm{def}}{=}\left(-1\right)^{n}\binom{-\omega}{n}=\binom{\omega-1+n}{n}$$ for all positive integers $...
MCS's user avatar
  • 1,284
2 votes
0 answers
135 views

Question regarding proof of Mertens' estimates in Montgomery-Vaughan's "Multiplicative number theory"

I have been trying to read Theorem 2.7 of Montgomery-Vaughan's "Multiplicative number theory" volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write $$\...
AK12N1's user avatar
  • 81
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
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
1 vote
0 answers
192 views

Prerequisites for Chen's theorem?

I am an undergraduate theoretical physics student, and I am trying to understand Chen's theorem. But when I tried to read Chen Jingrun's 1973 paper (https://www.sciengine.com/Math%20A0/doi/10.1360/...
Ben's user avatar
  • 11
1 vote
0 answers
112 views

If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?

Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
Vincent Granville's user avatar
1 vote
0 answers
97 views

Bounds and repulsion domains for the Dirichlet eta function $\eta(\sigma+it)$, for fixed $\sigma$

Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T =\Omega_T(\sigma)$ as the ...
Vincent Granville's user avatar
1 vote
0 answers
166 views

Euler product over subsets of primes

It is well known that $$\prod_p\,(1-p^{-1})=\frac 1 {\zeta(1)}=0$$ Given an arbitrary prime $\,q\,$ is it true that $$\prod_{q\,|\,p+1}\,(1-p^{-1})=0\;\;\;?$$ Thanks.
Augusto Santi'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
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
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
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
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
0 votes
0 answers
104 views

Validity of a Tauberian theorem for Dirichlet series

I encountered this statement about Dirichlet series but couldn't find a similar result in Korevaar's "Tauberian Theory". Is this statement valid? Statement: Let $f(s) = \sum_{n=1}^{\infty} \...
 Babar's user avatar
  • 611
0 votes
0 answers
88 views

Accentuating the appearance of convergence of the Möbius function Dirichlet series on the line $\sigma = \frac{2}{3}$ in the critical strip

Set the constant $c$ to: $$c = -\frac{3}{4}$$ which is in the interval: $$-1 < c < 0$$ and let the matrix $A$ be: $$A(n,k)=[k|n] - [n=k](1+c)$$ Then form the matrix power series: $$M=\sum _{n \...
Mats Granvik's user avatar
  • 1,183
0 votes
0 answers
101 views

Prime races in two competing arithmetic progressions - error bound

I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
Vincent Granville's user avatar
0 votes
0 answers
151 views

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 ...
Vincent Granville'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
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