All Questions
Tagged with dirichlet-series power-series
13
questions
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 ...
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 ...
- 43
1
vote
0
answers
28
views
Are there some natural bijections between general Dirichlet series and power series?
The theory of general Dirichlet series and the theory of power theory have some analogs:
The abscissa, line and half-plane of convergence of a Dirichlet series
are analogous to radius, boundary and ...
- 53
1
vote
2
answers
153
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Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$
By trying to prove that Riemann's Zeta function is analytically expendable to the whole plane with one pole, I went aside and noticed this identity about formal power series (which are obviously ...
- 1,697
0
votes
1
answer
187
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Dirichlet L functions [closed]
I would like to know more special Dirichlet L functions (like Zeta function for instace). Despite Zeta, Beta, Eta, Lambda and Hurwitz zeta are there more special Dirichlet L functions? I went to the ...
- 516
1
vote
0
answers
32
views
Generalization of convolution operation? [duplicate]
In integral transforms, convolution is defined by
$$
(f*g)(t)=\int_{-\infty}^\infty f(\tau)g(t-\tau)\mathrm d\tau
$$
satisfying the commutative, associative property, and
$$
\mathcal F\{f*g\}=\mathcal ...
- 7,193
1
vote
0
answers
55
views
Dirichlet Theorem Expansion to power series
I am nearly complete in my understanding of this beautiful theorem of Dirichlet. On page 34 of Davenports book (Multiplicative number theory) the following breakdown is present and I do not understand ...
- 81
1
vote
1
answer
83
views
Are all series in the elementary Ramanujan class R = 1 non-summable by analytic continuation of Dirichlet series?
We say that a series $\sum_{n=1}^\infty a_n$ and the corresponding power series $f(x)=\sum_{n=1}^\infty a_nx^n$ belong to the Ramanujan class $R=1$ if $g(x)=f(x)-f(x^2)$ is Abel summable at $x=1$ (...
user299632
2
votes
1
answer
354
views
Does the Abel sum 1 - 1 + 1 - 1 + ... = 1/2 imply $\eta(0)=1/2$?
If $\sum_{n=1}^\infty a_n$ is Abel summable to $A$, then necessarily $\sum_{n=1}^\infty a_n n^{-s}$ has a finite abscissa of convergence and can be analytically continued to a function $F(s)$ on a ...
user299632
0
votes
1
answer
79
views
What's the function that is related to 3 as the Riemann zeta function is related to 2?
For $f(x)=\sum_{n=0}^\infty a_n x^n$, a real number $R\neq 1$, $g(x)=f(x)-Rf(x^2)$ Abel summable at $x=1$, $g(1)=\lim_{x\to 1^-} g(x)$, the elementary Ramanujan sum of $f(x)$ at $x=1$ is defined by $f(...
user299632
1
vote
1
answer
119
views
Translations AND dilations of infinite series
Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ...
- 8,000
6
votes
1
answer
287
views
"Reduction of Dirichlet series into power series"
In a paper of Riemann, he states to following formal identity.
If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then
$$\Gamma(s)f(s)=\int\limits_{0}^{\...
- 3,967
4
votes
1
answer
213
views
A numeral system built around Dirichlet series, by analogy of how positional numeral systems are built around power series?
For any natural number and chosen base p, the number admits a unique expression of the form $a_np^n + ... + a_2p^2 + a_1p^1 + a_0$, where $a_k < p$ for all k. This property is effectively what ...
- 8,000