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 ...
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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 ...
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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 ...
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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 $(-1)^k$ ...
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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 ...
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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:
$$\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}=\int_1^\...
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1
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Perron's formula in the region of conditional convergence
I am a bit confused about the proof of Perron's formula. It states that for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ and real numbers $c > 0$, $c > \sigma_c$, $x > 0$ we have
$$\...
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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 ...
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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 ...
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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 $$\sum_{n\le x}\...
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Proof of $\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx$ and $\limsup_{x\to\infty} \frac{\log|A(x)|}{\log x} = \sigma_c$
Theorem. Let $A(x) := \sum_{n\le x} a_n$. If $\sigma_c < 0$, then $A(x)$ is a bounded function, and $$\sum_{n=1}^\infty a_n n^{-s} = s\int_1^\infty A(x) x^{-s-1}\, dx \tag{1}$$ for $\sigma > 0$. ...
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Gamma integral in Dirichlet L-series
I am studying Dirichlet L-series in Algebraic Number Theory by Neukirch (Chap VII, section 2). In order to define the completed L-series of a character $\chi$ it started considering the gamma integral ...
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A question about Landau’s theorem for Dirichlet series and integrals
A well known theorem of Landau’s for Dirichlet series and integrals goes as follows (I copy the theorem almost exactly as it appears in Ingham’s Distribution of Prime Numbers, Theorem H in Chapter V, ...
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Dirichlet series of an elementary function
Is there an example of an elementary function (different from Dirichlet polynomials, i.e. cutoff Dirichlet series) which has a know Dirichlet expansion (known coefficients)?
I am aware of the ...
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"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
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$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?
Let $p_n$ be the $n$ th prime number.
Let $f(s)$ be a Dirichlet series defined on the complex plane as :
$$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
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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, ...
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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 $A(x):=\sum_{n\le x}a_n$.
...
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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 ...
user1236730
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2
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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*}
...
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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 ...
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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 ...
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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 ...
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1
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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 ...
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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 $$\sum _{k=1}^{\infty } (-1)^{k+1}...
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Can we extend the Divisor Function $\sigma_s$ to $\mathbb{Q}$ by extending Ramanujan Sums $c_n$ to $\mathbb{Q}$?
It can be shown that the divisor function $\sigma_s(k)=\sum_{d\vert k} d^s$ defined for $k\in\mathbb{Z}^+$ can be expressed as a Dirichlet series with the Ramanujan sums $c_n(k):=\sum\limits_{m\in(\...
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How to construct a Dirichlet series that cannot be analytically continued beyond its abscissa of absolute convergence?
If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$.
Are ...
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How fast does the proportion guaranteed by dirichlet converge?
I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
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Residue of a Dirichlet Series at $s=1$
I have encountered this problem of determining the leading term in the Laurent expansion of a Dirichlet series. Let $d(n)$ be integers and consider the Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{...
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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\...
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