Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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Question on term of derived formula for $\log\zeta(s)$

The derived formula $$\log\zeta(s)=-\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n\in\mathbb{P}} \left(2 \tanh ^{-1}\left(1-2 n^s\right)-i \pi\right)\right),\quad s>1\tag{1}$$ ...
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Questions on summatory function related to non-integer-powers

Consider the summatory function $$f(x)=\sum\limits_{n=1}^x 1_{n\ne k^m}\tag{1}$$ where $1_{n\ne k^m}$ is the non-integer-power indicator function which returns $1$ when $n$ is a non-integer-power and $...
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1 vote
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On the abscissa of convergence of a Dirichlet series.

I am trying to find the abscissa of convergence of the Dirichlet series for the arithmetic function $|\mu(n)|$. I have managed to show that $$\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\...
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How to write a convergent function as a dirichlet expansion?

Normally when using Dirichlet series, it is used as a generating function to prove certain results like Dirichlet's theorem. I'm wondering whether it's possible to write a function $f(s)$ as a ...
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Variation for Dirichlet's test.

In the original statment of Dirichlet's test (and Abel's as well), we discuss improper integrals at unbounded intervals only (like [a, inf]). I wonder if there is a variation of the test for bounded ...
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Characteristic function of the mean of a Dirichlet process

In a 1984 paper discussing the characteristic function of the mean of a random distribution driven by a Dirichlet process ${\sf DP}(M,G_0)$ (Ferguson, 1973), $M>0$, Hajime Yamato sets a constraint ...
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Are there some natural bijections between general Dirichlet series and power series?

The theory of general Dirichlet series and the theory of power theory have some analogs: The abscissa, line and half-plane of convergence of a Dirichlet series are analogous to radius, boundary and ...
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2 votes
1 answer
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Does the L-series of a modular form that is not a cusp form make sense?

If $f$ is a modular form and we let $a_n$ be the Fourier coefficients, then the $L$-Series associated to $f$ is $$ L(s,f)=\sum_n\frac{a_n}{n^s} $$ Usually, we only define this for cusp forms that is ...
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1 answer
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Does the Euler product for the Rankin-Selberg convolution of two eigenforms require the Ramanujan conjecture?

Suppose $f$ and $g$ are weight $k$ eigenforms for the Hecke operators, normalized, with Fourier coefficients $a_{n}$ and $b_{n}$ respectively. I wanted to see if I could derive a Euler product for $$L(...
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1 answer
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Uniform convergence about Dirichlet integral $f(s):=\int_1^\infty\frac{a(x)}{x^s}\,dx =\lim\limits_{T\to\infty}\int_1^T\frac{a(x)}{x^s}\,dx$

On page 87 of Ingham's book: The Distribution Of Prime Numbers, the author asserts the following results, but does not give proof. Let $a(x)$ be a bounded and integrable function over any finite ...
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Multiplication of Dirichlet Generating Functions

Let $A(x)$ be a Dirichelet generating function given by: $A(x) = \frac{a_1}{1^x}+\frac{a_2}{2^x}+\frac{a_3}{3^x}+...=\sum_{n=1}^{\infty}\frac{a_n}{n^x}$. Given the Dirichelet generating functions $A(x)...
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Question on convergence of explicit fomulas for summatory functions related to Dirichlet series

Given the totient summatory function $$\Phi(x)=\sum\limits_{n=1}^x\varphi(n)\tag{1}$$ and the related Dirichlet series $$\frac{\zeta(s-1)}{\zeta(s)}=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{...
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How to prove that $\sqrt{3}\pi/6=\prod_{p \equiv 1 \pmod{6}} \frac{p}{p-1}\prod_{p \equiv 5 \pmod{6}} \frac{p}{p+1}$ with $p \in \mathbb{P}$?

I would like to prove the formula $$\frac{\sqrt{3}\pi}{6}=\left(\prod_{\substack{p \equiv 1 \pmod{6} \\ p \in \mathbb{P}}} \frac{p}{p-1}\right) \cdot \left(\prod_{\substack{p \equiv 5 \pmod{6} \\ p \...
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Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

This is a math competition problem for college students in Sichuan province, China. As the title, calculate the limit $$\lim_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}.$$ It is clear that the ...
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Dirichlet Series of Square Full Integers.

As in the title, I want to find the Dirichlet series $F$ of the indicator function for cube full integers $f(n)=1 \iff p^3|n, \forall p|n$ and $f(n)=0$ otherwise. Since $f$ is clearly multiplicative, $...
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3 votes
2 answers
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Upper bound for L-series of modular form from its integral representation

I've read that if $f$ is a cuspidal modular form (that's also an eigenfunction for the Hecke operators) for $SL_{2}(\mathbb{Z})$ of weight $k$, then its L-series $L(w,f)$ satisfies the bound $$L(w,f) &...
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Dirichlet series of a Gaussian

Let $a > 0$ and $s \in \mathbb{C}$. I'm wondering whether the series $$ \sum_{n=1}^\infty \dfrac{e^{-an^2}}{n^s} $$ are encountered in the literature? Specifically I'm interested in the following: ...
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proving that dirichlet series has non negative coefficients and does not converge for all $s\in\mathbb{C}$

given $Z(s)=\zeta^2(s)\zeta(s+it)\zeta(s-it)$ I need to prove that Z(s) is represented by a dirichlet series with non negative coefficients whiche does not converge for all $s\in\mathbb{C}$. I have ...
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5 votes
2 answers
116 views

Proof that the series $\sum_{n=2}^{\infty}\frac{[\Omega(n)]^\alpha}{n^2}$ converges

Let's consider the series $$f(\alpha)=\sum_{n\gt1}\frac{[\,\Omega(n)\,]^\alpha}{n^2}$$ where $\Omega(n)$ denotes the number of prime factors of $n$ counted with their multiplicity and $\alpha\ge0$ is ...
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1 answer
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Dirichlet series for $\zeta^3(s)/\zeta(2s)$.

I am currently studying number theory and our instructor refers to Apostol's book on Analytic number theory for the chapter Dirichlet series.In that book,there is an exercise which is as follows: Let $...
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Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}$.

I am a graduate student of Mathematics. I have started reading number theory. I encountered a problem of analytic number theory. Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{...
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1 vote
1 answer
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Necessary and sufficient condition to be completely multiplicative

I want to prove that $f*f=f \tau$ iff $f$ is completely multiplicative. The "if" part was relatively easy, using $f(g*h)=(fg)*(fh)$ and plug $g=h=1$ for all $n$. Juxtaposition is ordinary, ...
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1 answer
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Looking for table of special values of the Dirichlet $L$-function

For double checking calculations I made I'd like to find a table of values of $L(-1,\chi_D)$ for small positive fundamental discriminats $D$. It there a table somewhere in the internet? Where? With $\...
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1 vote
2 answers
117 views

Proving $\prod_{n=0}^{\infty}\left(1+\frac{x}{a^n}\right)=\sum_{n=0}^{\infty}\frac{(ax)^n}{\prod_{k=1}^{n}(a^k-1)}$

By trying to prove that Riemann's Zeta function is analytically expendable to the whole plane with one pole, I went aside and noticed this identity about formal power series (which are obviously ...
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Proof that Dirichlet series $\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2}$

So I want to prove the following: $$\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^2}=\frac{5}{2},$$ where $\omega(n)$ is the number of distinct prime factors of $n.$ I computed it to $10^{10}$ and it does ...
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1 vote
2 answers
127 views

Prove $\sum_{d | n} \mu(d) (\log(d))^2=0$ [duplicate]

If $n$ is a positive integer with more than 2 distinct prime factors, how to prove that $\sum_{d | n} \mu(d) (\log(d))^2=0$? I struggle on how to continue from this. Suppose $n=p_1 p_2 ... p_r$, ...
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1 vote
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Are there are other known functions that give the same set of zeros as the Riemann zeta function inside the critical strip besides $\eta(s)$?

The zeros of the Dirichlet's eta function $\eta(s)$ inside the critical strip match the non-trivial zeros of the Riemann zeta function $\zeta(s)$, as $\eta(s) = (1 - 2^{1-s}) \zeta(s)$. What are the ...
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1 answer
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If the Dirichlet's eta function is conditionally convergent in the critical strip how can we ever compute its values there?

I'm studying infinite series, but I'm a physicist, not a mathematician. I got it from Hardy's THE GENERAL THEORY OF DIRICHLET'S SERIES that the Dirichlet series $\eta(s) = 1^{-s} - 2^{-s} + 3^{-s} - \...
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Proof Riemann Zeta Series based on $\eta(s)$ has only one pole.

This proof is my understanding of a very interesting comment by @leoli1 on my previous related question about the following extended Riemann Zeta function which converges for $\sigma>0$ where $s=\...
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What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $?

What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $? In the lecture note our instructor claimed that the abscissa of convergence of the above ...
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A question about Dirichelet Series.

I am looking at some of the formulea here ... https://en.wikipedia.org/wiki/Dirichlet_series Show \begin{eqnarray*} \frac{ \zeta(s) \zeta(s-a) \zeta(s-2a)}{ \zeta(2s-2a) } = \sum_{n=1}^{\infty} \frac{...
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Variant of Möbius inversion: $b(n) = \sum_{d^2 \mid n} a(n/d^2) d^\alpha$

I'm trying to understand a step in a classic paper of Rankin. In Rankin's paper Contributions to the theory of Ramanujan's function $\tau(n)$ and similar arithmetical functions, he defines $$ b(n) := \...
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4 votes
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What do we know about the analytic continuations of Dirichlet series?

Let $s=\sigma+it$ be a complex number and define the function: $$F(s)=\sum_{k=2}^{\infty}\frac{p_\pi(k)}{k^s}$$ Where $p_\pi(k)$ is the number of unordered factorizations of $k$, corresponding to OEIS ...
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1 vote
0 answers
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Summary of Dirichlet Series Convergence from Apostol's IANT

I've been trying to learn about Dirichlet series, in particular from Apostol's IANT textbook. The textbooks tend to present result and not discuss them narratively, so I am left unsure of my correct ...
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1 vote
1 answer
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Step in Apostol's IANT on Dirichlet Series

I can't explain the following step in Apostol's IANT regarding Dirichlet Series. Specifically, how does the magnitude of the following $$\left | \int_a^b t^{s_0 -s -1} \right |$$ become the following?...
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2 votes
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Question related to potential closed-form representation of Catalan's constant

The motivation for this question is to find a closed-form representation for Catalan's constant. Formula (1) below for the Dirichlet beta function $\beta(s)$ (which I believe is globally convergent) ...
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4 votes
2 answers
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Prove that the infinite $\sum_{\text{ p prime}}\frac{1}{2^p}$ is an irrational number. [duplicate]

Prove that the infinite $\sum_{\text{ p prime}}\frac{1}{2^p}$ is an irrational number. My progress: Suppose $$\omega = \sum_{\text{ p prime}}\frac{1}{2^p}= \frac{1}{2^2}+\frac{1}{2^3}+\dots$$ Also ...
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Question on Difference Root representations of $\eta(2 n+1)$ and $\beta(2n)$ where $n\in\mathbb{N}$

I've noticed the Dirichlet eta function $\eta(s)$ and the Dirichlet beta function $\beta(s)$ can be represented by difference roots at odd and even positive integers respectfully. For example, ...
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3 votes
0 answers
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The average order of the divisor functions ${\sigma _\alpha }(n)$, where $\alpha < 0$ (Apostol, Intro to Analytic Number Theory, p.61)

In Apostol’s book, Theorem 3.6 (p.61) states a result concerning the average order of ${\sigma _\alpha }(n)$, where $\alpha < 0$ I am including an outline of Apostol’s approach, I hope I have ...
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1 answer
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Dirichlet transform of $e^{(2 \pi i / 3) \Omega(n)}$

The Dirichlet transform of the Liouville function $\lambda(n)$ is famously $$ \sum_{n=1} \frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)}\tag{1}$$ The Liouville function is defined by $$ \lambda(n) ...
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2 answers
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How to find the coefficent of a term in a Dirichlet generating function in Mathematica?

For a normal Dirichlet generating function like $Zeta[s]^2$, I can get the coefficient of the n-th term by applying Dirichlet convolution of the two constant functions. But how to find the coefficient ...
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1 answer
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For which $s$ does $\sum 1/p^s$ converge?

A well-known result is that $\sum 1/n^s$ converges for $\operatorname{Re}(s)>1$. Question: For which $s$ does $\sum 1/p^s$ converge, where $p$ is over all primes? Notes: Intuitively there are ...
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2 votes
1 answer
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Product over the primes with relation to the Dirichlet series

What is the value of $\displaystyle \prod_p\left(1+\frac{p^s}{(p^s-1)^2}\right)$ I got this product by defining a function $a(n)$ such that $a(n)=a(p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_n^{a_n})=a_1a_2a_3......
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2 votes
1 answer
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Confused by boundedness and convergence of Dirichlet Series (Apostol 11.6 Lemma 2, Theorem 11.8)

Apostol's IANT Section 11.6 is on "The half-plane of convergence of a Dirichlet Series". In it he proves that if a Dirichlet series is bounded at $s_0$ then it is also bounded at $\sigma>...
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1 vote
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Convergence of a Dirichlet series

For a fixed positive integer $j$, consider the arithmetical function : $$\vartheta _{j}(k+1)=\left\{\begin{matrix} 1 \;\;, & k+1=j^{l}\;\;(l=1,2,3...)\\ 0 \;\;, & \text{otherwise} \end{...
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1 answer
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Last step in Apostol's Section 11.6 Lemma 2 (Dirichlet Series)

Apostol's IANT Section 11.6 on the half-plane of convergence presents and proves a Lemma 2. Question: I can't understand the simplifications he does at the last step. The image below highlights the ...
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1 vote
1 answer
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Validity of proof showing difference in abscissa of convergence and absolute convergence of Dirichlet Series is $\leq1$?

The following is a step-by-step proof/derivation showing the difference in abscissae of convergence $a_c$ and absolute convergence $a_a$ is never more then 1. Question: Is this simple proof correct? ...
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Why is $\lim_{x\rightarrow \infty} \sum_{x<n\leq\infty}a_{n}n^{-s}= 0$ a sufficient condition for convergence?

Assume the following is true $$\left|\sum_{x_{1}<n\leq x_{2}}\frac{a_{n}}{n^{s}}\right| \leq Kx_{1}^{-\sigma}$$ where $s=\sigma+it$ and $a_n$ are complex, and all other variables are real, and $\...
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Why is Abel's Identity (Apostol Theorem 4.2) valid for complex functions?

Apostol uses the Abel Identity developed early in his book as Theorem 4.2 (image below) $$ \sum_{y<n\leq x}= A(x)f(x) - A(y)f(y) - \int_{y}^{x}A(t)f'(t) dt $$ to prove a result about complex ...
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  • 1,756
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0 answers
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Product of two Dirichlet series without absolute convergence.

This is from Stein and Shakarchi's Complex Analysis, Chapter 7, Exercise 2. Show that if $\{ a_m \}$ and $\{ b_k \}$ are two sequences of complex numbers with bounded partial sums, and $\Re(s) > 0$,...
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