Questions tagged [dirichlet-series]
For questions on Dirichlet series.
563
questions
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\...