Questions tagged [dirichlet-series]

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Exceptional zeros of a convolutional inverse

Let $\kappa:\lbrace 1,2,3,\ldots\rbrace\longrightarrow \mathbb Z$ be the convolutional inverse in the Dirichlet ring of $n\longmapsto {n+1\choose 2}$. It is defined by $\kappa(1)=1$ and by the ...
Roland Bacher's user avatar
2 votes
1 answer
129 views

Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?

I originally asked this question on Math StackExchange a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow. This Math ...
Steven Clark's user avatar
2 votes
2 answers
582 views

An inequality related to Catalan's constant and $\zeta(3)$

Problem : Show that : $$\frac{1}{\zeta(3)}<2C-1$$ Where we can see the zeta function and the Catalan's constant . After a bounty on Maths Stack Exchange there is no satisfying answer . See https://...
Erik Satie's user avatar
7 votes
2 answers
347 views

Proving a series for $\pi$ by Plouffe

Simon Plouffe found experimentally a series for $\pi$ that can be written as $$\frac{\pi}{24} = \sum_{n=1}^\infty \frac{1}{n} \left( \frac{3}{e^{\pi n}-1} -\frac{4}{e^{2\pi n}-1} +\frac{1}{e^{4\pi n}-...
Jaume Oliver Lafont's user avatar
0 votes
0 answers
94 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
4 votes
1 answer
216 views

Zeros of Dirichlet function $L(s,\chi_4)$

I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function $$ L_4^* (s,\chi_4)...
Vincent Granville's user avatar
3 votes
1 answer
186 views

Positivity of partial Dirichlet series for a quadratic character?

Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
Shan's user avatar
  • 71
-2 votes
1 answer
105 views

Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character

A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around. I define the function $$ L_4^*(s) = \...
Vincent Granville's user avatar
2 votes
1 answer
81 views

A sum related to the first moment of quadratic $L$-functions at $s=1$

Let $(\frac{m}{n})$ be the Jacobi quadratic symbol defined for positive squarefree odd integers $n,m$. Does the following sum go to infinity? $$ \sum_{1\leq n \leq (\log x)^{100} } \mu^2(2n) \sum_{(\...
Dr. Pi's user avatar
  • 2,862
1 vote
0 answers
125 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
154 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
  • 18.7k
6 votes
0 answers
169 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
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3 votes
0 answers
104 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
1 vote
0 answers
102 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
2 answers
261 views

Abscissa of convergence for a very specific Dirichlet series / Euler product

I am interested in the convergence of the following Euler product: $$ \prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}. $$ The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...
Vincent Granville's user avatar
1 vote
0 answers
89 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
4 votes
0 answers
53 views

Number of cosets by quaternion order

Let $D$ be a quaternion division algebra over $\mathbb Q$ and let $\Lambda\subset D$ be a maximal order. For $n\in\mathbb Z$ let $\Lambda^{(n)}$ denote the set of all $\lambda\in\Lambda$ with reduced ...
Echo's user avatar
  • 1,378
9 votes
1 answer
309 views

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 ...
Holomorphic manifold's user avatar
14 votes
1 answer
524 views

Mean value theorem for Dirichlet series - optimize?

Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality $$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq \sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$ ...
H A Helfgott's user avatar
  • 18.7k
12 votes
2 answers
497 views

Computation of modified Gauss sums

Let $\chi$ be a primitive Dirichlet character of conductor $q$. I want to compute numerically $$G(k)=\sum_{n\bmod q}\chi(n)e^{2\pi i n(n-k)/(2q)}$$ for all $k$ with $0\le k<2q$ with $k\equiv q\...
Henri Cohen's user avatar
  • 10.4k
0 votes
0 answers
128 views

Dirichlet series associated to divisor function

How can I express the following Dirichlet series $$\sum_{n=1}^{\infty}\frac{d_{k}^{2}(n)}{n^{s}}$$ (where $d(n)$-divisor function;$k\geq 1$) in terms of the zeta function?
user155294's user avatar
1 vote
0 answers
164 views

Zeta functions of schemes of finite type over $\mathbb{Z}$

Let $X$ be a scheme of finite type over $\mathbb{Z}$. In Section 11 of my 2008 paper in J. Number Theory, "Ring structures on groups of arithmetic functions," I define an additive analogue $...
Jesse Elliott's user avatar
0 votes
1 answer
56 views

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 ...
truebaran's user avatar
  • 8,748
5 votes
1 answer
154 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 ...
Mayank Pandey's user avatar
8 votes
0 answers
436 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 ...
Daniel Loughran's user avatar
1 vote
1 answer
228 views

Generalizing closed form representations related to conjectured analytic formulas for $f_a(x)=\sum\limits_{n=1}^x a(n)$

Consider the summatory function $f_a(x)$ defined in formula (1) below where the related Dirichlet series $F_a(s)$ defined in formula (2) below converges for $\Re(s)\ge 2$. $$f_a(x)=\sum\limits_{n=1}^...
Steven Clark's user avatar
6 votes
0 answers
591 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
0 votes
1 answer
437 views

Questions on analytic representations of the Kronecker delta function $\delta(x-1)$ and the Moebius function $\mu(n)$

This question is related to analytic formulas for $a(n)$ where $f_a(x)$ and $F_a(s)$ defined in formulas (1) and (2) below are the summatory function and Dirichlet series associated with $a(n)$. $$...
Steven Clark's user avatar
0 votes
0 answers
129 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
7 votes
0 answers
123 views

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, ...
davidlowryduda's user avatar
1 vote
0 answers
271 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 $w=1$. ...
Vincent Granville's user avatar
6 votes
4 answers
373 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 ...
Kim's user avatar
  • 3,974
2 votes
0 answers
98 views

An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series

Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...
Anton Devyatkov's user avatar
1 vote
1 answer
232 views

Continuing an analytic continuation of the Dirichlet $\eta$-function?

The Dirichlet $\eta$-function is defined as: $$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} \qquad \Re(s) > 0$$ and has the full analytical continuation: $$\eta(s) = \sum_{n=1}^N \frac{(-1)^{...
Agno's user avatar
  • 4,019
2 votes
1 answer
95 views

Are there theorems dealing with the "amount of oscillatory divergence" of series?

Are there a set of theorems dealing with "amount of divergence" series? Let me explain by example. The Dirchlet $\eta$ series $\sum_n (-1)^{n-1} n^{-x}$ converges when $x > 0$. We may say ...
Shree's user avatar
  • 203
3 votes
0 answers
70 views

Continuation of $\sum \sigma_\nu(n) a(n) n^{-s}$ for $a(\cdot)$ coming from a half-integral weight form

In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...
davidlowryduda's user avatar
2 votes
0 answers
175 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
  • 1,256
12 votes
1 answer
483 views

Convergence of the series involving Mobius functions $\sum_{k,d} \mu(d) x_{kd}$

(I originally asked this question here, but the problem appears much more difficult than I think after a moment of thought, so I think it might be more suitable to post it here. Please tell me if this ...
Ma Joad's user avatar
  • 1,469
2 votes
0 answers
126 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
160 views

Multiplicity of zeros of partial sums of the Dirichlet Eta function

I am studying ways to approach the problem of the multiplicity of zeros of the partial sums of the Dirichlet Eta functions: $$ \sum_{n=1}^{K}\frac{(-1)^{n-1}}{n^{s_o}} = 0 $$ more in particular, ...
Luca's user avatar
  • 362
1 vote
2 answers
198 views

What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomial if it is exists?

I have done some search many times on web to find any approximation of $\log|(\zeta'(s))|$ in Dirichlet polynomial but I didn't got it, Probably that $\log(|\zeta'(s)|$ dosn't have a Dirichlet ...
zeraoulia rafik's user avatar
2 votes
2 answers
546 views

Convergence of Euler product and Dirichlet series in the same half-plane?

I'm crossposting this from math.stackexchange because I think it might be inappropriately research-level for the community over there. Suppose we have an Euler product over the primes $$F(s) = \prod_{...
Rivers McForge's user avatar
2 votes
1 answer
246 views

Binomial transform of Dirichlet series

Let $\Theta(s)$ be a Dirichlet series , and let $\beta$ be its abscissa of convergence: $$\Theta(s)=\sum_{n=1}^{\infty}\frac{\theta(n)}{n^{s}}\;\;\;\;\;\;\Re(s)>\beta$$ And let $\left\{a_{n}\right\}...
mohammad-83's user avatar
9 votes
2 answers
512 views

Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$

I was working with some Dirichlet series and I realized that I have never seen any general conditions under which \begin{equation} \sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
Milo Moses's user avatar
  • 2,509
2 votes
1 answer
377 views

Moments of Dirichlet $L$-functions on the critical line

I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line, $$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$ where $\chi$ is a ...
Anurag Sahay's user avatar
1 vote
0 answers
147 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
2 votes
1 answer
170 views

Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
asrxiiviii's user avatar
0 votes
1 answer
249 views

Properties of Dirichlet series

I have a question about convergence and properties of Dirichlet series. it seems a bit interesting and different about the convergences of Dirichlet Series to me. With $c\in [0,1]$, $$f(n) = \pm 1,...
user1062's user avatar
  • 105
0 votes
2 answers
307 views

What is the dirichlet series of $f(n)=\sum_{d | n}(\log d) / d$ function? [closed]

My opinion is ; We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question. i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain $F(S)=\sum_{n=...
user1062's user avatar
  • 105
15 votes
2 answers
667 views

If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$?

Helmut Hasse has proved that for $s \in \mathbb{C}-\{1\}$ the Riemann zeta function can be written as: $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}...
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