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4 questions
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}\...
- 105k
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 $L(s,\chi)$...
- 105k
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 ...
- 105k
3
votes
1
answer
963
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 ...
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