# Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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### Roots of trancendental equations and their relation to Dirichlet series and Mellin transforms

Consider a function of the form $F(x)=x^{\alpha}f(\ln(x))$, with $0<\alpha<1$ and $c_1<f(\ln(x))<c_2$ for some positive constants $c_1,c_2$, such that $F(x)$ is strictly increasing. ...
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### sum of :$\sum_{k=1}^\infty\frac{(-1)^k}{2k-1} \cos(2k-1)$

How can I find the sum of :$$\sum_{k=1}^\infty\frac{(-1)^k}{2k-1} \cos(2k-1)$$ I don't fully understand the parseval identity so I am asking if we can use it to find the sum, and if so, how I should ...
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### What is meant by $\sum_{d \le x}f(d)$ in this equation?

Wikipedia's page (here) on the average order of arithmetic functions gives the following as a means of finding such an order using Dirichlet Series: Define $f$ as an arithmetic function on $n$, and ...
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### Proving the relation between the Dirichlet eta function and the Riemann zeta function [closed]

The problem I am trying to solve is: I need to prove the relation between the Dirichlet eta function and the Riemann zeta function $\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$. But I have no clue ...
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### Is it possible to express $\eta(\frac{1}{2})$ succinctly in terms of $\pi$ or some other constant(s)?

Dirichlet Eta function is defined by Dirichlet series $\eta(s) = \sum_n \frac{(-1)^{(n+1)}}{n^s}$, which converges for $\Re(s) > 0$. I calculated an approximate value for $\eta(\frac{1}{2})$ by ...
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### Is there a function whose Dirichlet series and whose Taylor series are the same?

That is, does there exist a sequence $\{a_n\}_{n\ge 1}\subset\mathbb{C}$, such that $$\sum_{n\ge 1} \frac{a_n}{n^z} = \sum_{n\ge 1} a_n (z-r)^n$$ for some $r\in\mathbb{C}$? Or perhaps shift there's ...
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### Abscissa (absolute) convergence of $f(n)=1 / \log ^{2}(2 n)$

i think about the convergence and absolute convergence of $f(n)=1 / \log ^{2}(2 n)$ . My idea is,it is open that $1/log ^{2}(2 n)$ is a nonnegative function, so its abscissa of convergence is equal ...
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### Problem of separation of variables for Dirichlet boundary data of Laplace's equation in polar coordinates

Need help here with figuring out boundary conditions for this problem. Also, for (i), I do know a general way or method but here I am confused since from both equations how do I find out my desired ...
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### Dirichlet Series in analytic number theory

I have a question about Abscissa of Convergence of Dirichlet series. The question is ; "Let $\sigma_{1}$ and $\sigma_{2}$ be real numbers with $\sigma_{1} \leq \sigma_{2} \leq \sigma_{1}+1 .$ ...
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### A series involving the Dirichlet Beta function; How to evaluate $\sum_{n=1}^\infty \frac{\beta(n)-1}{n}$?

Let the beta and the zeta function be defined as usual: \begin{align} & \beta(s) & = & \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^s} & = & 1-\frac{1}{3^s}+\frac{1}{5^s}\dots +\...
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### A question about Dirichlet series [closed]

i have the following question given 2 options as i) and ii) Let $f(n)$ be the unique positive real-valued arithmetic function that satisfies $\sum_{d | n} f(d) f(n / d)=1$ for all $n$ . (i.e., $f$ ...
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### Verifying the global maxima of a smooth function using Dirichlet Series?

Question Given $f$ is a smooth function and $b_r = \sum_{d \mid r} a_d\mu(\frac{r}{d})$ with $\lim_{n \to \infty} \frac{\log^2(n)}{n}\sum_{r=1}^n |b_r| = 0$. Then if (and only if) $f(k)$ is a global ...
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### Generalising Dirichlet Distribution to Dirichlet Process

I'm trying to follow a tutorial paper on generalizing Dirichlet Distribution Finite Mixture Models to Dirichlet Process Infinite Mixture Models; Li, Y., Schofield, E., & Gönen, M. (2019). A ...
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### Which functions do Dirichlet series represent?

I'm reading Serre's $\textit{A course in Arithmetic}$ where he defines a Dirichlet series to be an infinite sum of the form $$f(z) = \sum\limits_{n=1}^{\infty} a_ne^{-\lambda_nz}$$ where $\lambda_n$ ...
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### what is the Dirichlet series of $\Lambda*\mu$

we have the following $\Lambda=log*\mu$ hence what is the Dirichlet series for $\Lambda*\mu$ is there any closed expression
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### How to calculate $\sum_{n=1}^\infty\frac{3^{\omega(n)}\Omega(n)}{n^s}$?

For $s\in\mathbb C$ with say $\Re s>1$, how to write $$\sum_{n=1}^\infty\frac{3^{\omega(n)}\Omega(n)}{n^s}$$ in terms of the Riemann Zeta function (where $\omega$ is the number of prime factors ...
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### Does $\sum_{n=1}^\infty \frac{1}{\phi(n)^s}$ have a euler product?

Does $$\sum_{n=1}^\infty \frac{1}{\phi(n)^s}$$ have a euler product and functional equation? $\phi(n)$ is the euler phi function. Since $\phi(n)$ is multiplicative I think the series could have a ...
### How does one obtain an expression for the Dirichlet series $g(s, \theta) = \sum_{n=1}^{\infty} \frac{\cos(n \theta)}{n^{s}}$?
I would like to obtain an expression for the function $$g(s, \theta) = \sum_{n=1}^{\infty} \frac{\cos(n \theta)}{n^{s}} \qquad (\#).$$ Here is what I've tried so far: we know from the definition of ...