Questions tagged [dirichlet-series]
For questions on Dirichlet series.
563
questions
4
votes
1
answer
47
views
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 ...
5
votes
0
answers
101
views
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 ...
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 ...
1
vote
0
answers
32
views
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 $(-1)^k$ ...
0
votes
0
answers
37
views
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 ...
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:
$$\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}=\int_1^\...
2
votes
1
answer
45
views
Perron's formula in the region of conditional convergence
I am a bit confused about the proof of Perron's formula. It states that for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ and real numbers $c > 0$, $c > \sigma_c$, $x > 0$ we have
$$\...
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 ...
1
vote
0
answers
75
views
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 ...
6
votes
0
answers
194
views
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 $$\sum_{n\le x}\...
0
votes
1
answer
107
views
Proof of $\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx$ and $\limsup_{x\to\infty} \frac{\log|A(x)|}{\log x} = \sigma_c$
Theorem. Let $A(x) := \sum_{n\le x} a_n$. If $\sigma_c < 0$, then $A(x)$ is a bounded function, and $$\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx \tag{1}$$ for $\sigma > 0$. ...
1
vote
1
answer
53
views
Gamma integral in Dirichlet L-series
I am studying Dirichlet L-series in Algebraic Number Theory by Neukirch (Chap VII, section 2). In order to define the completed L-series of a character $\chi$ it started considering the gamma integral ...
2
votes
1
answer
88
views
A question about Landau’s theorem for Dirichlet series and integrals
A well known theorem of Landau’s for Dirichlet series and integrals goes as follows (I copy the theorem almost exactly as it appears in Ingham’s Distribution of Prime Numbers, Theorem H in Chapter V, ...
0
votes
1
answer
91
views
Dirichlet series of an elementary function
Is there an example of an elementary function (different from Dirichlet polynomials, i.e. cutoff Dirichlet series) which has a know Dirichlet expansion (known coefficients)?
I am aware of the ...
0
votes
0
answers
62
views
"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
0
votes
0
answers
20
views
$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?
Let $p_n$ be the $n$ th prime number.
Let $f(s)$ be a Dirichlet series defined on the complex plane as :
$$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
0
votes
0
answers
35
views
$ 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, ...
0
votes
1
answer
67
views
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 $A(x):=\sum_{n\le x}a_n$.
...
-1
votes
1
answer
279
views
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 ...
1
vote
2
answers
104
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*}
...
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 ...
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 ...
0
votes
0
answers
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 ...
1
vote
1
answer
47
views
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 ...
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 $$\sum _{k=1}^{\infty } (-1)^{k+1}...
4
votes
0
answers
74
views
Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
0
votes
0
answers
26
views
How to construct a Dirichlet series that cannot be analytically continued beyond its abscissa of absolute convergence?
If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$.
Are ...
0
votes
0
answers
15
views
How fast does the proportion guaranteed by dirichlet converge?
I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
2
votes
1
answer
113
views
Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
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\...
2
votes
0
answers
92
views
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 ...
0
votes
0
answers
89
views
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, $\sigma,t\in\mathbb{R}$:
$$H_{ t }^{(\sigma+it)}=\zeta (\sigma+it)-\zeta \
(\sigma+it, t ...
0
votes
0
answers
40
views
LCM sum with $\log $'s
If I want to evaluate $$\sum _{[r,r']\leq x}\log r\log r'$$ I could write it as an integral using Perron's formula, pick up a pole, and get a main term which involves looking at (the derivatives at $\...
0
votes
1
answer
99
views
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
1
vote
1
answer
99
views
Perron's Formula with $\rm{si}$-Remainder
I'm studying the book `Multiplicative Number Theory I. Classical Theory' by Hugh L. Montgomery and Robert C. Vaughan, and I don't understant a step of the proof for Perron's Formula(in Section 5.1) ...
3
votes
1
answer
63
views
The evaluation of the coefficient of the Dirichlet series $\zeta'(s)^2$
The derivative of Riemann zeta function is $\zeta'(s)=-\sum_{n=2}^{\infty}(\log{n}) n^{-s}.$
The square of $\zeta'$ is the following Dirichlet series:
$$\zeta'(s)^2=\sum_{n=4}^{\infty}a_nn^{-s},$$
...
3
votes
1
answer
238
views
Alternating Dirichlet series involving the Möbius function.
It is well known that:
$$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$
with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function.
Numerical ...
1
vote
1
answer
49
views
inequality involving two dirichlet series
Let $f\left( s \right) = \sum\limits_{n = 1}^{\infty}\left[ a_{n} \cdot \left( {\frac{1}{n^{s}}-\frac{1}{n^{1 - \operatorname{conj}\left( s \right)}}} \right) \right]$ and Let $g\left( s \right) = \...
1
vote
1
answer
78
views
How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.
Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
3
votes
1
answer
79
views
Convolution Method for Bound
I am reading A survey of gcd-sum functions where the following result is stated:
Let $P(n)$ be the Pillai's arithmetical function. The Dirichlet series of $P$ is given by:
$$\sum_{n=1}^\infty \frac{P(...
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
$$
\int_{0}^{\infty}\vartheta_2(...
6
votes
1
answer
154
views
Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$
I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$):
\begin{aligned}
\int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4
\text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
1
vote
0
answers
40
views
Question on conjectured method of extending convergence of Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$
The question here is motivated by this Math StackExchange question and this Math Overflow question which indicate the evaluation of the Dirchleta eta function
$$\eta(s)=\underset{K\to\infty}{\text{lim}...
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 $\...
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)...
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 ...
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 $L(\sigma,f)$ ...
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 ...
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 $\...
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 ...