# Tagged Questions

For questions on Dirichlet series.

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### Can the coefficients of a Dirichlet series be recovered?

Specifically if I have a known function $F(s)$ is there some way I can find a function $f(n)$ that satisfies this equation? $$F(s) = \sum_{n=1}^\infty \frac{f(n)}{n^s}$$ I'm imagining something ...
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### How do we evaluate this Dirichlet L-series

In this answer, David Speyer, whose answer is magnificent, states that "The sum $\sum \chi_3(n)/n$ is only slightly less well known; it is $\pi/(3 \sqrt{3})$.", where $\chi_3(n)$ is the character ...
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### Product of two series to get a series decomposition of zeta in the critical strip

$\def\sfrac#1#2{% \small#1% \kern-.05em\lower0.1ex/\kern-.025em% \lower0.4ex\small#2}$I've been working on gaining an intuitive understanding of the analytic continuation of the zeta ...
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### Uniform and ordinary convergence of a series.

I have a series of the kind $\sum\limits_{n=1}^{\infty} \frac{a_nt^n}{1+t^{2n}}$, $t\in(0,\infty)$. Btw, $a_n$ are real and they are determined by several integrals, but it is possible to compute any ...
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### Having some trouble using Dirichlet's test to show series convergence,

The series is $$\sum_{k=1}^{\infty} \frac{1}{k^\alpha}\log\left(1+\frac{1}{k}\right)$$ Since the summand is a product of two factors, and the log factor is monotonically decreasing to zero, as n goes ...
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### What is the use of Dirichlet Integral? [closed]

How can I find the value of $$\large\int_0^\infty\left(\dfrac{\sin x}x\right)^5dx$$ using Contour Integrals? I attempted it using Integration by Parts and got the an got the answer. I have studied ...
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### Partial GCD - Sum

$\sum\limits_{n = 2}^{M} \sum\limits_{m = 1}^{R} GCD(m,n)$ $R = (\lfloor\dfrac{N}{n}\rfloor-n) \% n$ $M = \lfloor\sqrt{N}\rfloor$ I calculated that - For N=10 the sum is 1 For N = 100 the ...
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### Multiplicative function and Euler product

Theorem 1 Let $G \subset \mathbb{C}$ be an area and $\sum_{n=1}^{\infty}a_n e^{-\lambda _ns}$ and $\sum_{n=1}^{\infty}b_n e^{-\lambda _ns}$ two Dirichlet-series that converge on $G$ and represents ...
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### Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?

I've done the following exercise: Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent? My approach: We're going to use the Dirichlet's ...
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### Dirichlet series expansion for Zeta(s)

Wikipedia lists a series expansion for $\zeta(s)$ here. How is the Dirichlet series below derived? I apologize in advance if this is a very simple question, I don't know much about Dirichlet series. ...
### Integral evaluation for the Riesz means special case $s_n=1$
At the moment, I am investigating the Riesz means defined as the series $$s^{\delta}(\lambda)=\sum_{n\leq\lambda}\left( 1-\frac{n}{\lambda} \right)^{\delta}s_n.$$ Consider the special case $s_n=1$. ...
### Prove or disprove the existence of f(k) when $\sum_{n=1}^m{\frac{a_n}{n^s}}=\sum_{n=1}^m\frac{a_n+f^n(x)}{n^s}$.
Can we find a function, $f(k)$, such that $$\frac{a_n}{n^s}+\frac{a_{n+1}}{(n+1)^s}+\frac{a_{n+2}}{(n+2)^s}=\frac{a_n+x}{n^s}+\frac{a_{n+1}+f(x)}{(n+1)^s}+\frac{a_{n+2}+f(f(x))}{(n+2)^s}$$ for some $x$...