# Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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### $L(1,\chi)$ exact value where $\chi$ is a non-principal Dirichlet character mod 3

If $\chi$ is a Dirichlet character, one defines its Dirichlet L-series by $L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}$. Let $\chi$ be a non-principal Dirichlet character mod 3. What is the ...
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### Picking the start of convergence $c$ for an integral expression of the Dirichlet $\eta$-function. What happens when $c \in \mathbb{C}$?

Using the following expression for the Dirichlet $\eta(s)$-function: $$\normalsize \eta(s,c) = -\frac{1}{\Gamma(s-c)} \int_0^{\infty} x^{s-c-1}\,\text{Li}_c(-\text{e} ^{-x}) \mathop{dx}$$ one could ...
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### How does this identity so elegantly combine an infinite sum in $\eta$ and an improper integral in $\Gamma$?

This is all well and good, but where did this come from? In the article on the Gamma function, Wikipedia shows most of its alternate definitions with clear proofs, yet in the article on the Dirichlet ...
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### Does my odd proof for the Abel sum for $\eta(-2)$ work?

EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" ...
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### Where does $s-1$ come from in this $\zeta(s)$ equation?

I have been working my way through this Arxiv paper concerning the analytic continuation of the zeta function. I don't understand the first equality in equation (19), page 6. In equation (11), the ...
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### Does the Dirichlet series for $\frac{\zeta(s+1)}{\zeta(s)}$ converge for $s>0$ as well as $\Re(s)>\sigma_c>0$?

This question pertains to the Fundamental Theorem of Dirichlet series which is stated on Wikipedia as follows (where $s=\sigma+i\,t$): There are now three possibilities regarding the convergence of ...
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### $\int_{0}^{\pi} D_{n}(y)dy=\frac{1}{2}$ Dirichlet

I need to calculate that $\int_{0}^{\pi} D_{n}(y)dy=\frac{1}{2}$ with $D_{n}(y)= \frac{1}{2\pi}\frac{\sin((n+\frac{1}{2})y)}{\sin(\frac{y}{2})}$ from Dirichlet. Now I tried to do this with the known ...
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### Analytical continuation for polylogarithm

It is known that the series presentation $$L_s(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^s}$$ for the polylogarithm is valid only in the open disk $|z|<1$. Outside this region, the polylogarithm is ...
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### Dirichlet series for square root of Riemann Zeta function

Can we obtain Dirichlet series for the function $\sqrt{\zeta(s)}$? Is it possible via Euler product for $\zeta(s)$?
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### Property of Dirichlet series which have $\sigma_{a}=1$

If the Dirichlet series $$\sum_{n=1}^{\infty}\frac{f(n)}{n^s}$$ converges absolutely for $\Re(s)>1$, does it follow that the Dirichlet series $$\sum_{n=1}^{\infty}\frac{f(n^2)}{n^s}$$ also ...
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### Function $\epsilon(s, \chi)$ related to Dirichlet L -function

I am trying some exercises in number theory from Tom M Apostol as my instructor doesn't gives any assignment. I am struck on this particular problem. ( Problem 12.9 on page 274). Can you please tell ...
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### Binomial transform of Dirichlet series

First off, i appologise for the long question, but it seems this is the only way i can convey my thoughts. Referring to this unanswered question on MO, i have thought for some time about it, and came ...
### What is the limit of this $a_r = e^{i \theta/r}$ Dirichlet Series?
Consider the following Dirichlet Series: $$D(s) = e^{i \theta} + \frac{e^{i \theta/2}}{2^s} + \frac{e^{i \theta / 3}}{3^s} + \dots$$ Is there a nice limit for the below?  \lim_{s \to 1} \frac{D(s)}{...