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Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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How to find the $\zeta$ representation of a $L$-series

Consider the following problem: Show that for $s>1$: $$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.$$ ($\mu$ denotes the Mobius function) My approach: One may first note that the ...
NTc5's user avatar
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Zeta Lerch function. Proof of functional equation.

so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following. In the article "Note sur la function" by Mr. Mathias Lerch, a ...
Nightmare Integral's user avatar
9 votes
3 answers
2k views

How does Wolfram Alpha know this closed form?

I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function: $$ \zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3). $$ How does WA know ...
Klangen's user avatar
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2 concise tables of “usual” series (mostly trigonometrics) and of "usual" L-series (Zeta, Eta, Beta...)

CONTEXT Common series are usually described as infinite sums, written as consecutive terms ending with (…). Or they can be described using the $\sum_{}$ symbol and arguments usually including $(-1)^k$ ...
olivierlambert's user avatar
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Building the theoretical foundation for generating functions - formal power series

I have read several documents on generating functions. I would like to inquire about two issues: Among the materials I have read, some mention generating functions constructed from formal power ...
Math_fun2006's user avatar
2 votes
1 answer
74 views

Dirichlet series and Laplace transform

Let $\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}$ be a Dirichlet series. It can be represented as a Riemann-Stieltjes integral as follows: $$\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}=\int_1^\...
Mateo Andrés Manosalva Amaris's user avatar
2 votes
1 answer
45 views

Perron's formula in the region of conditional convergence

I am a bit confused about the proof of Perron's formula. It states that for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ and real numbers $c > 0$, $c > \sigma_c$, $x > 0$ we have $$\...
Manuel Eberl's user avatar
3 votes
1 answer
128 views

Power series for $\sum_{n=0}^\infty(-1)^n/n!^s$ (around $s=0$)

I'm looking for ways to compute the coefficients of the power series $$ \sum_{n=0}^\infty\frac{(-1)^n}{n!^s}=\sum_{k=0}^{\infty}c_k s^k $$ (a prior version of the question asked whether such an ...
metamorphy's user avatar
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A question about Lemma 15.1 (Landau’s theorem for integrals) in Montgomery-Vaughan’s book

Lemma 15.1 in Montgomery-Vaughan’s analytic number theory book is Landau’s theorem for integrals. My question is, why is it necessary to have $A(x)$ bounded on every interval $[1,X]$? Doesn’t the ...
EGME's user avatar
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Proof of Theorem 1.1 of Analytic Number Theory by Iwaniec & Kowalski

I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski. They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\...
Kangyeon Moon's user avatar
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Proof of $\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx$ and $\limsup_{x\to\infty} \frac{\log|A(x)|}{\log x} = \sigma_c$

Theorem. Let $A(x) := \sum_{n\le x} a_n$. If $\sigma_c < 0$, then $A(x)$ is a bounded function, and $$\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx \tag{1}$$ for $\sigma > 0$. ...
stoic-santiago's user avatar
1 vote
1 answer
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Gamma integral in Dirichlet L-series

I am studying Dirichlet L-series in Algebraic Number Theory by Neukirch (Chap VII, section 2). In order to define the completed L-series of a character $\chi$ it started considering the gamma integral ...
Mario's user avatar
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A question about Landau’s theorem for Dirichlet series and integrals

A well known theorem of Landau’s for Dirichlet series and integrals goes as follows (I copy the theorem almost exactly as it appears in Ingham’s Distribution of Prime Numbers, Theorem H in Chapter V, ...
EGME's user avatar
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1 answer
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Dirichlet series of an elementary function

Is there an example of an elementary function (different from Dirichlet polynomials, i.e. cutoff Dirichlet series) which has a know Dirichlet expansion (known coefficients)? I am aware of the ...
F. Jatpil's user avatar
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"Mollifier" of the Dirichlet L-function

I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
djangounchained0716's user avatar
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0 answers
20 views

$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?

Let $p_n$ be the $n$ th prime number. Let $f(s)$ be a Dirichlet series defined on the complex plane as : $$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
mick's user avatar
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$ 0 = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)} \implies Re(s) \leq \frac{1}{2}$?

Define $f(s)$ as $$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)}$$ where we take the upper complex plane as everywhere analytic. Notice this is an antiderivative of the Riemann Zeta function, ...
mick's user avatar
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Asymptotics for the number of $n\le x$ which can be written as the sum of two squares. Is Perron's formula applicable?

For all $n\ge 1$, let $$ a_n = \begin{cases} 1\quad&\text{if $n$ can be written as the sum of two squares;}\\ 0&\text{otherwise} \end{cases} $$ I am interested in $A(x):=\sum_{n\le x}a_n$. ...
Mastrem's user avatar
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-1 votes
1 answer
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Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]

I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ...
user avatar
1 vote
2 answers
102 views

Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$ for $\chi$ mod $5$

My HW question is: Evaluate the series $$L(1, \chi_5) = \sum_{n=1}^\infty \frac{\chi_5(n)}{n},$$ where $\chi_5$ is the unique nontrivial Dirichlet character mod $5$. My work is: \begin{align*} ...
Clyde Kertzer's user avatar
2 votes
1 answer
116 views

Evaluate $L(1, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n}$ for $\chi$ mod $3$

Here is the homework question I am working on: Evaluate (as a real number) the series $$L(1, \chi_3) = \sum_{n=1}^\infty \frac{\chi_3(n)}{n},$$ where $\chi_3$ is the unique nontrivial Dirichlet ...
Clyde Kertzer's user avatar
2 votes
1 answer
273 views

Dirichlet series with infinitely many zeros

Can a Dirichlet series have infinitely many zeros and be nonzero? To be precise, by a Dirichlet series I mean a function of the form $s\mapsto \sum_{n\geq 1}\frac{a_n}{n^s}$ where the domain is the ...
Croqueta's user avatar
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55 views

How do we increase the region of convergence for the Riemann Zeta function (using Dirichlet Series form)?

The Riemann Zeta Function can be defined as: $\zeta(s)=\sum \frac 1 {n^s}$ for $s>1$. The series converges for $s>1$. wiki (https://en.wikipedia.org/wiki/Riemann_zeta_function) mentions that: An ...
stack.tarandeep's user avatar
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1 answer
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Manipulating Dirichlet series generating functions

This is from p.$61$ in Wilf's "generatingfunctionology" As a step to solving for the $b$'s in terms of the $a$'s Given: $a_n = \sum_{d\mid n}b_d$ Consider the Dirichlet power series ...
No infinity's user avatar
0 votes
1 answer
44 views

For what values of $c$ is $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}=0$ when $x=\exp \left(-\frac{\rho _1}{c}\right)$?

The alternating Dirichlet series, the Dirichlet eta function, can be written in the form: $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}$ For what values of $c$ is $$\sum _{k=1}^{\infty } (-1)^{k+1}...
Mats Granvik's user avatar
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4 votes
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Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?

It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
K. Makabre's user avatar
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0 votes
0 answers
26 views

How to construct a Dirichlet series that cannot be analytically continued beyond its abscissa of absolute convergence?

If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$. Are ...
isekaijin's user avatar
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0 votes
0 answers
15 views

How fast does the proportion guaranteed by dirichlet converge?

I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
Bruno Andrades's user avatar
2 votes
1 answer
113 views

Residue of a Dirichlet Series at $s=1$

I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
Gabrielle Rodriguez's user avatar
1 vote
0 answers
61 views

Asymptotic order of the square of the modulus of the second derivative of the Dirichlet kernel in zero

Consider the Dirichlet kernel $D_N(x)=\sum_{|k|\le N} e^{ikx}$. Its second derivative reads as $$D_N^{\prime\prime}(x) = -\sum_{|k|\le N} e^{ikx}k^2.$$ What is the asymptotic order of $|D_N^{\prime\...
James 's user avatar
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2 votes
0 answers
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Dirichlet series solution to Poisson Point Process question

Reposted to MathOverflow because the bounty on this post expired, with no solutions or comments received. For any discrete subset $S$ of $\mathbb{R}^d$, consider a digraph formed by placing an edge ...
Jim Ferry's user avatar
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0 votes
0 answers
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About the definition of generalized harmonic numbers and an identity

Some software packages make use of the following definition for generalized harmonic numbers. In what follows, $\sigma,t\in\mathbb{R}$: $$H_{ t }^{(\sigma+it)}=\zeta (\sigma+it)-\zeta \ (\sigma+it, t ...
EGME's user avatar
  • 405
0 votes
0 answers
40 views

LCM sum with $\log $'s

If I want to evaluate $$\sum _{[r,r']\leq x}\log r\log r'$$ I could write it as an integral using Perron's formula, pick up a pole, and get a main term which involves looking at (the derivatives at $\...
tomos's user avatar
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0 votes
1 answer
99 views

can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?

can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$? if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
Haidara's user avatar
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1 vote
1 answer
99 views

Perron's Formula with $\rm{si}$-Remainder

I'm studying the book `Multiplicative Number Theory I. Classical Theory' by Hugh L. Montgomery and Robert C. Vaughan, and I don't understant a step of the proof for Perron's Formula(in Section 5.1) ...
Kangyeon Moon's user avatar
3 votes
1 answer
63 views

The evaluation of the coefficient of the Dirichlet series $\zeta'(s)^2$

The derivative of Riemann zeta function is $\zeta'(s)=-\sum_{n=2}^{\infty}(\log{n}) n^{-s}.$ The square of $\zeta'$ is the following Dirichlet series: $$\zeta'(s)^2=\sum_{n=4}^{\infty}a_nn^{-s},$$ ...
FFGG's user avatar
  • 982
3 votes
1 answer
238 views

Alternating Dirichlet series involving the Möbius function.

It is well known that: $$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$ with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function. Numerical ...
Agno's user avatar
  • 3,191
1 vote
1 answer
49 views

inequality involving two dirichlet series

Let $f\left( s \right) = \sum\limits_{n = 1}^{\infty}\left[ a_{n} \cdot \left( {\frac{1}{n^{s}}-\frac{1}{n^{1 - \operatorname{conj}\left( s \right)}}} \right) \right]$ and Let $g\left( s \right) = \...
haidara gams's user avatar
1 vote
1 answer
78 views

How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.

Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
matt stokes's user avatar
3 votes
1 answer
79 views

Convolution Method for Bound

I am reading A survey of gcd-sum functions where the following result is stated: Let $P(n)$ be the Pillai's arithmetical function. The Dirichlet series of $P$ is given by: $$\sum_{n=1}^\infty \frac{P(...
Juan Esteban Arevalo Gomez's user avatar
12 votes
1 answer
1k views

Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$

The paper mentioned a proposition: $$ \int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }. $$ Its equivalent is $$ \int_{0}^{\infty}\vartheta_2(...
Setness Ramesory's user avatar
6 votes
1 answer
154 views

Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$

I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$): \begin{aligned} \int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4 \text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
Setness Ramesory's user avatar
1 vote
0 answers
40 views

Question on conjectured method of extending convergence of Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$

The question here is motivated by this Math StackExchange question and this Math Overflow question which indicate the evaluation of the Dirchleta eta function $$\eta(s)=\underset{K\to\infty}{\text{lim}...
Steven Clark's user avatar
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1 vote
1 answer
85 views

What’s the best bound on the Dirichlet coefficients of $\zeta(s-1)^2/\zeta(s)$

We have $\frac{\zeta(s-1)^2}{\zeta(s)} = \sum\limits_{n\ge 1} \frac{a_n}{n^s}$, where $a_n = \sum\limits_{d|n} \mu(d) \sigma_0(\frac{n}{d}) \frac{n}{d} = \sum\limits_{d|n} \phi(d) \frac{n}{d}$. Here $\...
Vik78's user avatar
  • 3,887
4 votes
1 answer
161 views

What is the value of $L'(1,\chi)$ where $\chi$ is the non-principal Dirichlet character modulo 4?

I was trying to compute the following sum: $$\sum_{n\le x}{\frac{r_2(n)}{n}}$$ where $r_2(n)=\vert\{(a,b)\in\mathbb{Z}^2:a^2+b^2=n\}\vert$. Using Abel's summation formula with $a_n=r_2(n)$, $\varphi(t)...
Desco's user avatar
  • 298
1 vote
0 answers
59 views

Proof that ring of formal Dirichlet series is isomorphic to a ring of formal power series over countably many variables

I found this article of E.D. Cashwell and C.J. Everett "The ring of number-theoretic functions" and they said Dirichlet series ring is isomorphic to formal power series ring of countably ...
toxic's user avatar
  • 329
0 votes
2 answers
42 views

Given a Dirichlet series that diverges, are there conditions to know when the modulus goes off to infinity?

I was working on a problem, and I had made the assumption that given a Dirichlet series $$ L(s,f)=\sum_{n\geq 1}\frac{f(n)}{n^s} $$ If I have some $\sigma\in\mathbb{C}$ such that $L(\sigma,f)$ ...
Steven Creech's user avatar
0 votes
1 answer
109 views

Dirichlet series for $\frac{\zeta(1-s)}{\zeta(s)}$ [closed]

Wikipedia (here) says that $\frac{\zeta(s-1)}{\zeta(s)}= \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^{s}}$ where $\varphi(n)$ is the totient function. Similarly, is there a known expression involving a ...
gregory's user avatar
  • 113
1 vote
1 answer
81 views

Convergence of sums in $\ell^p \implies \ell^{p-\epsilon}$

Supose $\displaystyle(b_n)_{n \in \mathbb{N}}$ is a sequence of positive real numbers that $$\displaystyle\sum_{n \in \mathbb{N}}(b_n)^{2} <\infty.$$ Does exists some $\epsilon>0$ such that $\...
Igor Soares's user avatar
0 votes
1 answer
49 views

What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?

The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by $$ A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p $$ is this serie calculated ...
Es-said En-naoui's user avatar

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