All Questions
Tagged with analytic-number-theory dirichlet-series
109 questions
5
votes
1
answer
693
views
An application of Mobius Inversion in a paper of Shintani
I've been reading about Shintani zeta functions and in particular with respect to finding the density of cubic discriminants as in the theorem of Davenport-Heilbronn. In Shintani's paper "On zeta-...
- 1,570
3
votes
1
answer
1k
views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of $...
- 534
3
votes
1
answer
846
views
Multiplicative functions whose Dirichlet series have essential singularities
What can be said about the partial sums of a complex-valued completely multiplicative function, let's say bounded by 1 in absolute value, if its Dirichlet series has an essential singularity?
As a ...
- 1,671
4
votes
2
answers
1k
views
Some Dirichlet series questions.
I asked this question on m.SE in an attempt to find out the right words to say for these questions I am about to ask.
In his great answer, Matthew Emerton explained that (cuspidal) automorphic L-...
9
votes
2
answers
1k
views
Analytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1
A sequence $a_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a_{mn} = a_m a_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum_n a_n n^{-s}$, absolutely convergent for $\Re s > 1$. If in ...
- 265
2
votes
1
answer
422
views
Dirichlet L series and integrals
If $f : t \to e^{-xt}$ with $x \geqslant 1$, and $d_n$ is the number of positive integers that divide $n$, I can show that
$$ \lim_{\epsilon \to 0^+} \sum_{n\geqslant 1}\frac{\epsilon^2 d_n f(\...
- 2,829
3
votes
2
answers
625
views
Continuation up to zero of a Dirichlet series with bounded coefficients
Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a ...
- 7,442
1
vote
1
answer
277
views
Shifted Dirichlet series
If $\sum_{n=1}^\infty \frac{a_n}{n^s} $ converges, does
$\sum_{n=1}^\infty \frac{a_n}{(n+1)^s} $ also converge?
- 2,302
8
votes
2
answers
1k
views
Is the maximum domain to which a Dirichlet series can be continued always a halfplane?
Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (...
- 7,836