Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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40 views

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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71 views

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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1answer
51 views

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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2answers
20 views

Proving identity using Dirichlet L functions

I'm trying to prove the following identity using Dirichlet L functions : ${\displaystyle \sum _{d\mid n}\varphi (d)=n}$ I have shown proved that the Dirichlet Series of $\varphi (n)$ equals to ${\...
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1answer
29 views

Prove that the convergence of the sum of its coefficients implies the convergence of the Dirichlet series

Suppose $\sum_{n=1}^\infty f(n)$ converges to a constant $A$, then prove that the Dirichlet series with coefficients $f(n)$ also converges if $\sigma>0$. This can be proved easily if the series $\...
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1answer
36 views

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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45 views

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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39 views

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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47 views

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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1answer
25 views

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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106 views

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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50 views

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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1answer
142 views

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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33 views

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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22 views

Why can we bound $L'(\sigma,\chi)$ by $O(\log^{2}q)$?

Pretty much as stated above. In Davenport's "multiplicative number theory" concerning Dirichlet L-functions, it says "We can easily prove that $|L'(\sigma,\chi)|=O(\log^{2}q)$ for $1-\frac{1}{\log q}\...
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2answers
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The length of the third piece in the Stick breaking process

I'm reading Nonparametric Bayesian Statistics Part I: some classical results. In page 14 and section 3.2.1 I can't understand why the length of the stick after the second break is: $$(1-Y_1)(1-Y_2)$...
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1answer
57 views

Establish a lower bound for the generalized alternating harmonic series

I'm given the following series:$$\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^p}$$ I need to show that the sum is greater than $1/2$ for every $p > 0$. For $p \ge1$ this is obvious, as it follows by ...
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25 views

Proper interpretation of Dirichlet probability distribution parameters and sample variance

I'm considering using the Dirichlet distribution to simulate a stochastic rock-breakage process and don't quite understand the interpretation of the concentration parameters ($\alpha_i$). [The choice ...
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21 views

Dirichlet characters acting on reduced residue classes/residue classes

In proving Dirichlets theorem on arithmetic progressions, we talk about the haracter $\chi$ defined as: 1: $\chi$ is periodic with period $q$. $\chi(n) = \chi(n+q)$ 2: $\chi(a) = 0$ if $(a,q)\...
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35 views

Cauchy Integral Formula and Dirichlet Series

Can someone explain to me this step from Harold Davenports 'Multiplicative Number Theory'? \begin{equation} \psi(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^{s}} \end{equation} Since $\psi(s)$ is ...
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22 views

Dirichlet series of real primitive character with two small real zeros

Let $\chi$ be a real primitive character with conductor $q$. I need to show that if the corresponding Dirichlet series $L(s,\chi)$ has two real zeros $\beta_0 \leq \beta_1 <1$, then $\beta_1 < 1 ...
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28 views

Dirichlet Theorem Expansion to power series

I am nearly complete in my understanding of this beautiful theorem of Dirichlet. On page 34 of Davenports book (Multiplicative number theory) the following breakdown is present and I do not understand ...
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22 views

Dirichlet-Multinomial Model Maximum Likelihood Estimation

Given a Dirichlet Prior $p_1...p_k$ ~ Dir($\alpha_1 ,..., \alpha_k)$ where $\alpha_1, ... ,\alpha_k = \alpha$ and given $X_1,..., X_n$ ~ Multinomial(n, $p_1,...,p_k$) what is the MLE estimator of ...
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22 views

Estimation of the alpha in a Multinomial-Dirichlet Model

I have a question regarding the Multinomial-Dirichlet model. Given a Dirichlet Prior, and a Multinomial likelihood. How can I estimate alpha (the parameter of the dirichlet) through a MLE in closed ...
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18 views

DP Chinese Restaurant Process Metaphor - Base Distribution?

On Wikipedia a Dirichlet Process $\text{DP}$ is described by the procedure; Draw a distribution $P$ from $\text{DP}(\alpha, H)$ Draw observations $X_1, X_2, \dots$ independently from $P$ where $\...
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43 views

Non-vanishing of Dirichlet $L$-function $L(s,\chi)$ for $\Re(s)=1$ [duplicate]

I know that if $\chi$ is a non-principal Dirichlet character then the $L$-function $L(s,\chi)$ doesn't vanish for $s=1$. But, how about $s=1+it$ with $t\neq 0$? I found in this post: Zeros of ...
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42 views

Question on the coefficient of the Dirichlet series related to $\frac{\zeta(s+2)}{\zeta(s)}$

This question is about the evaluation of $a(n)$ defined in (1) below which is related to the Riemann zeta function $\zeta(s)$ as illustrated in (2) below. (1) $\quad a(n)=\sum\limits_{d|n}\frac{\mu\...
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1answer
41 views

Dirichlet series - distance between $\sigma_u$ and $\sigma_c$

In a book available here we see the following In summary the above says: if a Dirichlet series is convergent at $s_0$, it is uniformly convergent in a compact sector right of it. Also if the ...
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1answer
20 views

$\sum n!a_n/(x(x+1)\cdots(x+n))$ and $\sum a_n/n^x$ has the same domain of convergence.

I $\sum n!a_n/(x(x+1)\cdots(x+n)), x\neq 0,-1,-2,\cdots$ and II $\sum a_n/n^x$ has the same domain of convergence. That is to say, if at $x$, $I$ converges, then $II$ converges. and vice versus.
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1answer
55 views

How is the parity condition determined for modular forms corresponding to $L(s, \chi_1) L(s, \chi_2)$?

This answer to a related question on Math StackExchange indicates the following: "This leads you to the conclusion that products like $\zeta(s)^2$ or $L(s, \chi_1) L(s, \chi_2)$ should correspond to ...
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1answer
74 views

Question on Relationship between Modular Forms and Dirichlet L-Series

The Wolfram MathWorld article Weisstein, Eric W. "Dirichlet L-Series." From MathWorld--A Wolfram Web Resource. mentions Hecke (1936) found a remarkable connection between each modular form with ...
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1answer
47 views

Questions on explicit formulas related to $\frac{L_{k,j}(s)}{\zeta(s)}$

This question is related to explicit formulas for $f_{k,j}(x)$ defined in (1) below where $\chi_{k,j}(n)$ is a non-principal Dirichlet character. (1) $\quad f_{k,j}(x)=\sum\limits_{n=1}^x a_{k,j}(n)\,...
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58 views

Is there a simple correction to this definition of the conductor of a Dirichlet character?

The website LMFDB Dirichlet Characters defines the conductor of a Dirichlet character as follows: The conductor of a Dirichlet character $\chi$ modulo $q$ is the least positive integer $q_1$ ...
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1answer
57 views

Is there a valid explicit formula for $f(x)=\sum\limits_{n=1}^x \frac{1}{n}\sum\limits_{d|n} \mu(d)\,d$?

This question is related to the function $f(x)$ defined in (1) below where A023900(n) is the Dirichlet inverse of Euler totient function $\phi(n)$. I believe the related Dirichlet series illustrated ...
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152 views

An entire function interpolating $\mu(n)$

This is in order to repair the pdf and answers of this user. $$f(x)=2\sum_{k\ge 0}\frac{x^{2k+1}}{\zeta(2k+2)}=2x\sum_{n\ge 1} \frac{\mu(n)/n^2}{1-x^2/n^2}, \qquad |x|<1$$ The RHS extends ...
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What is the variation in Dirichlet density at x for subsets i mod 5 (i=1,2,3,4)?

x being a (large) integer. The density is approximately the same for every i=1,2,3,4. Is there a formula for the variation in Dirichlet densities for the subsets? I am looking for a limitation of the ...
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2answers
83 views

Value of infinite sum $\sum_{n=1}^\infty \frac{\sin n}{\sqrt{n}}$?

The series $\sum_{n=1}^\infty \frac{\sin n }{\sqrt{n}}$ is clearly convergence as can be shown with the Dirichlet's test. But what is this value and how to evaluate this sum to get a closed form ...
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67 views

Bounding Integral Involving Riemann Zeta

For $2 \le T \le x$, can we bound $$\int_1^{1+\frac{1}{log x}}\frac{x^s}{s}\zeta(\sigma + iT) \, \mathrm{d}\sigma \ll \frac{x \log^3 x}{T}$$ Question Background We get the expression for $Z(s)$ ...
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38 views

Why is $\zeta_m(s)=\prod\limits_\chi L(s,\chi).$ a Dirichlet series with positive coefficients?

I know from this link: $\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients why $\zeta_m(s)$ is a Dirichlet ...
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23 views

Can the following be represented by a Fourier series : cos^-1(sin(2𝑥)) and |(sin(x))^(-1/2)|?

I am referring to the book K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press, 2000. Dirichlet conditions may be summarised by the ...
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1answer
62 views

Trouble Proving $\sum_{n=1}^\infty \frac{\mathrm{d}(n)^2}{n^s}=\frac{\zeta(s)^4}{\zeta(2s)}$

I am running into considerable trouble trying to prove the identity in the question. I figure the solution will come from Euler-products, so here was my attempt. I want to show that $$ \sum_{n=1}^\...
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1answer
68 views

Find Dirichlet series of $2^n$

How can I find the Dirichlet series of $2^n$? The Dirichlet series of a sequence $\{a_n\}_{n=1}^\infty$ is defined as $f(s) = \sum_{i = 1}^\infty \frac{a_n}{n^s}$. If $\{a_n\}_{n=1}^\infty$ is ...
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1answer
68 views

Finding Dirichlet series of $C_n^2$ and $2^\frac{n}{d}$

How can I find the Dirichlet generating function of $C_n^2$ $n\geq1$ and $2^\frac{n}{d}$ , $n\geq1$ , $d|n$. I tried a lot of time to do both of these , setting the series in the formula : $\...
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1answer
30 views

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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2answers
107 views

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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21 views

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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1answer
126 views

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 ...
3
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1answer
103 views

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 ...
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0answers
41 views

On a proof of a lemma by Titchmarsh/Bohr and Landau

In the Titchmarsh treatise "The Theory of the Riemann zeta function", at page 159 of the first edition, we have a lemma at paragraph 8.7. I joined an image My question is about the proof of the lemma ...
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3answers
94 views

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 ...

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