For questions on Dirichlet series.

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4
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2answers
55 views

Showing $\sum\frac{\sin(nx)}{n}$ converges pointwise

I do not understand how one can say using "Dirichlet conditions" that $\sum_{n=1}^{\infty}\dfrac{\sin(nx)}{x}$ is pointwise convergent. I know the proof for $x=1$ but how can one show it is convergent ...
1
vote
0answers
12 views

Can you give an example of an irreducible element of the ring of Dirichlet series with integer coefficients?

According to this. The ring of Dirichlet series with integer coefficients is a UFD. Can you give an example of an irreducible element in that ring?
2
votes
2answers
42 views

Dirichlet $L$ functions at $s=2$

Let $\chi$ be a Dirichlet character and let $L(\chi,s)$ denotes its Dirichlet $L$-function. What is the value of $L(2,\chi)$ ? Or simply, is $L(2,\chi)/\pi^2$ rational ? Many thanks for your answer ...
0
votes
1answer
30 views

Writing Dirichlet series in infinite product.

In Serre's $A \, Course\, In \,Arithmetic$, it says the following: $\sum\limits_{n=1}^{\infty}c(n)/n^s= \prod\limits_{p \,\rm prime}\frac{1}{1-c(p)p^{-s}+p^{2k-1-2s}}$ $\Longleftrightarrow$ ...
1
vote
1answer
33 views

$\zeta_m(s)=\prod\limits_{p\nmid m} \frac{1}{\left(1-\frac{1}{p^{f(p)s}}\right)^{g(p)}}$ is a Dirichlet series with non-negative coefficients

Let $p$ be a prime number, $m$ be any integer, $f(p)$ be the order of $p$ in $(Z/mZ)^*$, $i.e.$ $p^{f(p)} \equiv 1 \pmod m$ with $f(p)$ smallest. Let $g(p)=\frac{\phi(m)}{f(p)}$ is a integer where ...
0
votes
0answers
16 views

Dirichlet Series of weakly multiplicative characters

Let $$ \sum_{n=1}^{\infty} a(n)n^{-s} = \prod_{p}\left((1-\alpha_{p}p^{-s})(1-\alpha_{p}'p^{-s})\right)^{-1},$$ where $ a(n) $ is weakly multiplicative (i.e $ a(n)a(m) = a(n,m) $ if $ ...
2
votes
1answer
64 views

Integral of a Dirichlet Series

I'm stuck at a problem of an exercise list... I'd like some help to solve it :) The problem: Suppose that the Dirichlet Series $$A(s)=\lim_{N \to \infty}\sum_{n=1}^Na(n)n^{-s}$$ has abscissa of ...
4
votes
0answers
28 views

Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...
1
vote
1answer
99 views

Prove that this sequence of integers is on average equal to zero.

Consider the sequence $\{a(n)\}_{n\in\mathbb{N}^*}$ that is defined by the Dirichlet series: $$\zeta (s)^2\cdot\left(1-\frac{1}{2^{s-1}}-\frac{1}{3^{s-1}}+\frac{1}{6^{s-1}}\right)=\sum_{n\geq ...
7
votes
3answers
203 views

How to prove $(\frac{1}{5^3}-\frac{1}{7^3})+(\frac{1}{11^3}-\frac{1}{13^3})+(\frac{1}{17^3}-\frac{1}{19^3})+…=(1-\frac{\pi ^3}{18\sqrt{3}})$

How to prove $$ \sum_{k=1}^\infty \left[\frac{1}{(6k-1)^3} - \frac{1}{(6k+1)^3}\right] = 1 - \frac{\pi^3}{18\sqrt{3}}$$ I think this equality likes the Dirichlet Beta function. The numerical value ...
1
vote
0answers
77 views

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 ...
1
vote
1answer
155 views

Abscissa of convergence for a Dirichlet series

Let $\alpha \in \mathbb{Z}$ and $f(n) = n^{i \alpha n}$. What is the abscissa of convergence, $\sigma_c$, for the associated Dirichlet series, $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$? Since $|f(n)| = ...
0
votes
0answers
34 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
3
votes
1answer
96 views

Special case of prime number theorem for arithmetic progressions 4k+1

In terms of the proof of prime number theorem for arithmetic progressions, I have seen many proofs involving with the concept of "character". Is there an alternative way (without such a concept) to ...
1
vote
0answers
46 views

Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
2
votes
1answer
45 views

express the dirichlet series for the sequence d(n)^2 in terms of riemann zeta.

Prove that $$\sum_{n=1}^\infty d(n)^2n^{-s}=\zeta(s)^4/\zeta(2s)$$ for $\sigma>1$ what i did: I already proved this formally, that is, without considering convergence. I use euler products, ...
10
votes
2answers
191 views

Calculating the abscissa of convergence for general Dirichlet Series

I'm currently interested in proving this theorem which I have been thinking for quite a while: Define a Dirichlet Series $$\sum_{k=1}^{\infty}a_k e^{-\lambda_k z}$$ where $\lambda_k$ is a strictly ...
5
votes
2answers
111 views

Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

For some time I've been playing with this kind of sums, for example I was able to find that $$ \frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right) $$ where $$ ...
0
votes
0answers
45 views

Dirichlet L series estimation

let $\chi$ be a non-principal character modulo $q$, $M\geq 1$. I have to prove that, if $\vert \sigma - 1 \vert \leq \frac{1}{\log M}$, then $\vert \sum_{n=1}^M \chi (n)n^{-s}\vert\leq 1+e\log M$ and ...
0
votes
0answers
14 views

Can this matrix be said to consist of Dirichlet characters?

When: $j=1$ the following formula: $$T(n,k)=\prod\limits_{m=1}_{m \mid n}^{\text{n}} \left(\exp ^{-\mu \left(\frac{n}{m}\right)}\left(\Lambda \left(\frac{n}{m}\right)\right) \chi _{\exp ^{-\mu ...
2
votes
1answer
67 views

Uniform convergence

How would you show that $$f(x)=\sum_{n=2}^\infty \frac{\sin(2\pi n x)}{n\log n}$$ converges uniformly on $x\in[0,1]$. The pointwise convergence can be proved by Drichlet test
0
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0answers
41 views

Plotting Dirichlet Series $L_{\tau}(s)$ related to Ramanujan tau function in Mathematica

According to wikipedia, Ramanujan tau function $\tau:\mathbb{N}\to\mathbb{Z}$ is defined by equating the coefficients of the power series on both sides of the identity: ...
0
votes
2answers
54 views

Why is $\mu \star E =e $ , where $\star$ denotes the Dirichlet Convolution operator?

Let $$ E(n) = 1 \qquad \forall n \in \mathbb{Z} $$ be the constant function, and let $\mu$ be the Möbius function. Based on the following definition of the latter function, where $\mu(n) = 1$ for ...
0
votes
1answer
39 views

A converse theorem for Hecke modular forms?

Let $f$ be a function which can be represented by a Dirichlet serie (for $\Re s$ big enough) $$f(s)=\sum_{n=1}^{\infty}{\frac{a(n)}{n^s}}$$ and which can be meromorphically extend to $\mathbb{C}$ by ...
0
votes
0answers
7 views

The Dirichlet G.F of $2^\Omega$

Let $\Omega(n)$ and $\omega(n)$ be the number of prime factors of $n$ and of distinct prime factors of $n$, respectively. The Dirichlet G.F of $2^\omega$ is well known, and I was wondering if that of ...
1
vote
1answer
70 views

Examples of divergent series summed by means of the analytic continuation of the corresponding

For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic. Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means ...
0
votes
2answers
60 views

What does “Choose N ~ Poisson(ξ), Choose θ ~ Dir ( α )” mean in the context of Latent Dirichlet Allocation

I'm reading http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ03.pdf and trying to understand the notation and concepts behind LDA, in order to implement it myself. I've followed some ...
2
votes
1answer
103 views

Dirichlet series and Riemann zeta function

Im trying to show, for $\Re(s)>1$, that $\displaystyle\sum_{n=0}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}$, where $d(n)= |\{k \mid k|n \}|$, number of positive integers that ...
0
votes
0answers
20 views

The inverse of a Dirichlet serie

When we consider a Dirichlet serie : $$F(s)=\sum_{n=1}^{+\infty}{\frac{f(n)}{n^s}}$$ which converges absolutly in the half-plane $\{ \Re(s) > \sigma \}$ with $f(1) \neq 0$, what can we say about ...
2
votes
0answers
46 views

Growth rate of arithmetical function

I'm interested in how one would estimate the growth rate of $$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$ I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head ...
0
votes
0answers
31 views

A functional equation for a Dirichlet serie

I'm looking for a functional equation for the following Dirichlet serie $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ? Thank you !
0
votes
0answers
37 views

Convolution Dirichlet series convergence

After theorem 4.1 in http://www.math.uiuc.edu/~hildebr/ant/main4.pdf, which says (paraphrased) Let $f$ and $g$ be arithmetic functions whose Dirichlet series converge absolutely in $s \in ...
1
vote
0answers
19 views

Is $\min \deg$ in a dirichlet subring interesting or is it always $1$?

Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for ...
3
votes
3answers
94 views

Inverse of Dirichlet series equality

I stumbled across a formula in here and tried to prove it for myself: $$\frac{1}{L(s,\chi)}=\sum\limits_{n=1}^{\infty}\frac{\mu(n)\chi(n)}{n^s}$$ However I got stuck. In my attempt I tried to show ...
2
votes
1answer
57 views

Product of zeta and its conjugate

Suppose we have the zeta function $\zeta(s)$, and we want to multiply it by its complex conjugate $\zeta(s)^*$. Since $\zeta(s)^* = \zeta(s^*)$, we get $\displaystyle \zeta(s)\cdot\zeta(s)^* = ...
4
votes
1answer
68 views

Motivation for using $L(1,\chi)$ in the proof of Dirichlet's Theorem

Having read the proof of Dirichlet's Theorem on the infinitude of primes in arithmetic progressions, I am left wondering what his motivation for studying $L(1,\chi)$ was and why it is reasonable that ...
0
votes
0answers
58 views

Generalized Riemann Hypothesis : Zeros of Dirichlet L function and its functional equation

Let $\chi$ pe a primitive character modulo q with $\chi(-1)=1$ ; L is the Dirichlet - L function Define, $\xi(z,\chi)=(q/\pi)^{z/2}\Gamma(z/2)L(z,\chi)$ Show that $L(z,\chi)$ has infinitely many ...
13
votes
1answer
711 views

Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
1
vote
0answers
35 views

Determing sequence from its Dirichlet series

Suppose I know the Dirchlet series $$\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \frac{\zeta(s)}{\zeta(3s)},$$ where $\zeta(s)$ is the usual Riemann zeta function. My question is - is there a way to ...
2
votes
3answers
374 views

how to show that this complex series converge?

If $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}$$ Converges( s is real) and $\operatorname{Re}(z)>s$. Then $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{z}}$$ also converges. $a_n$ is complex sequence.
0
votes
2answers
35 views

Define this Function on [0,1]

I need a function, $f$ that is bounded but not integrable on [0,1], but $f^2$. I am thinking I should modify the Dirichlet function and use it to create a function whose square is constant. I am stuck ...
3
votes
1answer
51 views

“Reduction of Dirichlet series into power series”

In a paper of Riemann, he states to following formal identity. If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then ...
2
votes
1answer
62 views

Evaluating Dirichlet series

It is well known that $$\eta(s)=\sum\limits_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^s} =(1-2^{1-s})\zeta(s)$$ But I have the wider problem of evaluating the following ...
1
vote
1answer
69 views

Dirichlet kernel identity

My question is about Dirichlet kernel identity. Why does: $$\sum_{k=-n}^{n}e^{ikx}=1+ 2\sum_{k=1}^{n}\cos(kx)$$
1
vote
1answer
369 views

How to make Dirichlet character table modulo $5$

There are four reduced residue classes $\mod 5$, namely $1, 2, 3, 4$ and thus four Dirichlet characters $\mod 5$ since $\phi(5)=4$. I understand how to deduce that the character can be $1$ or ...
1
vote
1answer
88 views

Convergence of Trigonometric Dirichlet series

Can it be proved that the following series converges for some integer value of $s$? $$\sum_{n=1}^\infty\frac{1}{n^s|\sin(n)|}$$ If so what value(s) of $s$ would it converge for?
8
votes
3answers
330 views

Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
1
vote
0answers
36 views

generalization of dirichlet series to integrals?

if there are Dirichlet series like $$ F(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s}} $$ why isn't an anlogue like $$ F(s)= \int_{1}^{\infty}dx \frac{a(x)}{x^{s}} $$ for example the generalization of ...
3
votes
1answer
81 views

Properties of Arithmetic Functions

I was recently working on arithmetic functions and using Perron's formula to obtain asymptotic estimates. One observation I made was that the Dirichlet series often can be written in terms of the ...
4
votes
1answer
105 views

Dirichlet series experiment - computing the rational coefficient

Let consider the sequence of numbers $a_n = 0,1,-1,0,1,-1,0,1,-1, ...$ extended periodically ( so it has period $9$, $a_{n+10}=a_n$. In fact, this is a Dirichlet character $a_n = \chi_9(n)$ modulo 9. ...