All Questions
Tagged with dirichlet-series analysis
15
questions
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 $\sigma > 0$. ...
- 14.2k
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 ...
- 558
2
votes
1
answer
122
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Is there any relationship between a Dirichlet series and the same series with the sequence "shifted" by one term?
Suppose
$$ F(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}} $$
is a Dirichlet series for the sequence $a_{1}, a_{2}, \ldots\in\mathbb{C}$. Then let
$$ G(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{(n+1)^{s}}...
- 2,461
20
votes
5
answers
770
views
Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$
This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit
$$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$
It is clear that the ...
- 915
1
vote
1
answer
53
views
$F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and $G(s)=\sum_{n=1}^\infty \frac{g(n)}{n^s}$ s.t $F(s_k)=G(s_k)$ show that $f(n)=g(n)$
Let $F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and $G(s)=\sum_{n=1}^\infty \frac{g(n)}{n^s}$ be two Dirichlet series which are absolutely convergent for $\Re(s)>a$ for some $a\in \Bbb R$. If there ...
- 9,098
1
vote
1
answer
42
views
Dirichlet series $\sum a_n n^{-x}$ where $(a_1+\dots+a_n)/n \rightarrow A$
I am working on some Dirichlet series and I am familiar with the proofs of convergence of the functions $\zeta$ and $\eta$.
And I was wondering if $g(x)=\sum a_n n^{-x}$ was normally convergent on $[a,...
- 9,427
1
vote
0
answers
3k
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What is the limit of this Dirichlet series?
Background & Motivation
I'm trying to verify/disprove the conjectured formula of the weighted integral of $f(x)$: The Definite Integral Problem (with a twist)?
$$ \lim_{k \to \infty} \lim_{n \...
- 1,177
4
votes
4
answers
453
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Is this class of series all demonstrably transcendental?
Question:
For a vector with integer entries $[a_0, a_1, \dots, a_{k-1}]$ is it true that when $\sum_{n=1}^\infty{\frac{a_{n-1 \mod k}}{n}}$ is not divergent it limits to some transcendental number ...
- 3,813
4
votes
0
answers
2k
views
Proving convergence of the Dirichlet Eta function
I've been struggling to prove this for a project as I'm not an expert in the field, so i decided to go back to basics.
The Dirichlet Eta Function is defined by:
$$ \eta (s) = \sum_{n=1}^{\infty} \...
- 523
2
votes
2
answers
1k
views
Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)
On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula:
$\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$
where $s>1$, $\chi$ is a ...
- 1,055
0
votes
1
answer
86
views
We can suppose without loss... General Dirichlet question
I would like to know why I can start supposing that $\sigma_{0}=0$ in the proof of the following theorem about Dirichlet series:
"Let $a_{n}\ge0$ for every $n\ge1$. Then the point $s=\sigma_{0}$ is a ...
- 13
3
votes
0
answers
47
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Integral evaluation for the Riesz means special case $s_n=1$
At the moment, I am investigating the Riesz means defined as the series
$$
s^{\delta}(\lambda)=\sum_{n\leq\lambda}\left( 1-\frac{n}{\lambda} \right)^{\delta}s_n.
$$
Consider the special case $s_n=1$. ...
user230715
2
votes
1
answer
272
views
Integral of a Dirichlet Series
I'm stuck at a problem of an exercise list... I'd like some help to solve it :)
The problem: Suppose that the Dirichlet Series
$$A(s)=\lim_{N \to \infty}\sum_{n=1}^Na(n)n^{-s}$$
has abscissa of ...
- 1,122
11
votes
2
answers
305
views
Interesting phenomenon with the $\zeta(3)$ series
I noticed that if one takes certain partial sums of the series for $\zeta(3)$:
$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$
an interesting phenomenon occurs ...
- 19.8k
2
votes
2
answers
320
views
Convergence of the series $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$
Could you help me to understand for which $x$ this series converge $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?
- 1,285