All Questions
7 questions
12
votes
1
answer
617
views
Convergence of the series involving Mobius functions $\sum_{k,d} \mu(d) x_{kd}$
(I originally asked this question here, but the problem appears much more difficult than I think after a moment of thought, so I think it might be more suitable to post it here. Please tell me if this ...
- 1,755
11
votes
3
answers
866
views
Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$
I was working with some Dirichlet series and I realized that I have never seen any general conditions under which
\begin{equation}
\sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
- 2,902
3
votes
1
answer
964
views
Abscissa of absolute convergence of the product of two Dirichlet series
I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
2
votes
2
answers
741
views
Convergence of Euler product and Dirichlet series in the same half-plane?
I'm crossposting this from math.stackexchange because I think it might be inappropriately research-level for the community over there.
Suppose we have an Euler product over the primes
$$F(s) = \prod_{...
- 263
1
vote
0
answers
112
views
If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?
Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
- 3,098
0
votes
0
answers
88
views
Accentuating the appearance of convergence of the Möbius function Dirichlet series on the line $\sigma = \frac{2}{3}$ in the critical strip
Set the constant $c$ to:
$$c = -\frac{3}{4}$$
which is in the interval: $$-1 < c < 0$$
and let the matrix $A$ be:
$$A(n,k)=[k|n] - [n=k](1+c)$$
Then form the matrix power series:
$$M=\sum _{n \...
- 1,183
0
votes
1
answer
369
views
Properties of Dirichlet series
I have a question about convergence and properties of Dirichlet series. it seems a bit interesting and different about the convergences of Dirichlet Series to me.
With $c\in [0,1]$,
$$f(n) = \pm 1,...
- 105