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Questions tagged [dirichlet-series]

For questions on Dirichlet series.

In mathematics, a Dirichlet series is any series of the form $$ \sum_{n=1}^{\infty} \frac{a_n}{n^s}, $$ where $s$ and $a_n$ are complex numbers and $n = 1, 2, 3, \dots$ . It is a special case of general Dirichlet series.

Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann $\zeta$ function is a Dirichlet series with $a_n=1$, as also are the Dirichlet $L$-functions.

It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Johann Peter Gustav Lejeune Dirichlet.

558 questions
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Dirichlet series, abscissa of absolute convergence $\neq$ abscissa of uniform convergence

It is well known that Dirichlet series, series of the form $$\sum_{n=1}^{\infty}\dfrac{a_n}{n^s},$$ where $\{a_n\}$ is a complex sequence and s is a complex variable, converge in half planes. The example…
Mathitis
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Dirichlet series' help

If $f(n)$ is an arithmetic function with $|f(n)|=1$, and $$\lim_{s\to+1} (s-1)\sum_{n=1}^\infty\frac{f(n)}{n^s}=0$$ Can I deduce that $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$ I tried abel summation but I am still having…
Ethan Splaver
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How to extract coefficients from Dirichlet series.

I know, there was a similar question like this: Can the coefficients of a Dirichlet series be recovered? But i can see that in case when given function(series) has only positive real numbers as a domain, then Perron's formula is useless here. Could…
mkultra
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Is there a known example of a Dirichlet L series having at least one multiple root?

The Davenport-Heilbronn function doesn't have a Dirichlet series since its a linear combination of two L functions. I mention that because its a common myth that this function is an L function.
crow
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2 question about dirichlet series generating functions

my question is , is there a sequence so we have the Dirichlet series $$ \frac{\zeta(s+1/2)}{\zeta(s)}= \sum_{n=1}^{\infty} \frac{a(n)}{n^s} $$ and the second is, given the dirichlet series for the division function $$ \zeta (s) \zeta(s-a)…
Jose Garcia
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Determining the coefficients of the reciprocal of a Dirichlet series

Given a Dirichlet series with coefficients $$ F(s)= \sum_{n=1}^{\infty}\frac{a(n)}{n^{s}},$$ is it then always possible (and how) to obtain the $ b(n) $ coefficients related to its reciprocal $$…
Jose Garcia
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Normalizing constant in Dirichlet distribution

According to references (e.g. Wikipedia and elsewhere), the Dirichlet distribution, parametrized by $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_K)$, is $$ D(x_1, \ldots, x_K) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i -…
Richard Hartley
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Dirichlet series of the divisor function

I want to show that the Dirichlet series of the divisor function $\sigma_k$ converges absolutly. I have found this: $$\sum_{n=1}^{\infty}\frac{\sigma_{k}(n)}{n^{s}}=\zeta(s)\zeta(s-k)$$ for all $s\in\mathbb{C}$ with $\text{Re}(s)>k+1$. I thought:…
user365151
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Which one will give one

We know that the odd terms of Dirichelt Beta function are $\frac{\pi }{4}$,$\frac{\pi^3 }{32}$,$\frac{5\pi^5 }{1536}$,... If we use the WolframAlpha to find the limit of the odd terms only divided by $n$ when the $N\rightarrow \infty $ as shown in…
E.H.E
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Property on average orders of multiplicative functions

I read the statement that, for a multiplicative function $h(n)$ we have $$\sum_{n
Wirdspan
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Boundedness of a Dirichlet series on vertical lines in $\mathbb{C}$

Proposition: Let $\left\{ a_{n}\right\} _{n\geq1}$ be a sequence of complex numbers, let: $$A\left(N\right)=\sum_{n=1}^{N}a_{n}$$ and suppose that there is a real number $c>0$ so that $\left|A\left(N\right)\right|=O\left(N^{c}\ln N\right)$…
MCS
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Does there exist a database for aDirichlet L-series?

I want to find an L-function with a Dirichlet character $\chi$(or with a product of zeta function) to be equal the following series $$ G(s)=\sum_{n >1 }\frac{1}{n^s}=1+\frac{1}{4^s}+\frac{1}{6^s}+\frac{1}{8^s}+......$$ Where $n $ is not prime I…
Aster Phoenix
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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 tutorial on Dirichlet process mixture modeling. Journal of…
aaronsnoswell
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Euler Product and Dirichlet Series

I know the proof of this statement already exist, i need a link regarding the proof. The statement goes as:Assume that the dirichlet series attached to f, i.e the series $$\sum_{n=1}^\infty \frac{f(n)}{n^s},$$ converge absolutely for all $r>r_0$. If…
dazai osamu
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Singularities of ordinary Dirichlet series

Is there an example of an ordinary Dirichlet series such that (a) the Dirichlet series diverges to infinity at the real point (R > 0) on the line of convergence, and (b) R is not a pole of the function represented by the Dirichlet series.
nickkatz2018
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