For questions on Dirichlet series.

learn more… | top users | synonyms

0
votes
2answers
27 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 ...
1
vote
1answer
52 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 ...
2
votes
0answers
35 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...
0
votes
0answers
14 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 ...
1
vote
0answers
61 views

Are even Dirichlet series constants?

I consider a Dirichlet series with an absolute abscissa of convergence $\sigma$ that can be meromorphically extend to $\mathbb{C}$: $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ and the analytic ...
2
votes
0answers
31 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
28 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
28 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
64 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
47 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
44 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
39 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 ...
11
votes
1answer
549 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
34 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
276 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
31 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
39 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
0answers
21 views

Field automorphism of $\mathbb{C}$ and Dirichlet series [closed]

Suppose $\sigma$ is a field automorphism of $\mathbb{C}$ such that for all complex number $s$ such that $\Re(s)>1$ one has ...
2
votes
1answer
51 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
45 views

Dirichlet kernel identity

My question is about Dirichlet kernel identity. Why does: $$\sum_{k=-n}^{n}e^{ikx}=1+ 2\sum_{k=1}^{n}\cos(kx)$$
0
votes
1answer
135 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
81 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?
7
votes
0answers
195 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
30 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
72 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
86 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. ...
0
votes
0answers
17 views

Triangle Mesh as a topological disk [duplicate]

I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My ...
0
votes
1answer
88 views

Dirichlet triangle mesh

I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My ...
0
votes
1answer
59 views

Proof of Dirichlet's Principle using Cauchy-Schwarz [closed]

I'm having real difficulty solving this problem below. I was wondering if anyone had any ideas about how to proceed? Thanks, Moehringer
1
vote
1answer
102 views

How to find the eigenvalue and eigenfunction of Laplacian?

Define a bounded domain $\Omega=(0,a)\times(0,b)$ What is the eigenvalue and eigenfunction of the Laplacian with homogeneous boundary condition? my first thought is something like $sin(n\pi ...
0
votes
0answers
10 views

Confusion related to intractability of a problem

I was reading this tutorial related to topic models and I have a confusion $p(\theta|\alpha)$ is a Dirichlet distribution $p(z_n|\theta)$ is a multinomial distribution $p(w_n|z_n)$ is a multinomial ...
4
votes
1answer
81 views

Continuous version of the Möbius inversion theorem

Is there a continuous version of Möbius Inversion. Essentially, using integrals instead of sums.
0
votes
1answer
33 views

How to construct a minimizing sequence?

Let ${u_k}$ be a harmonic function sequence that is continuous on a unit disk. How to construct the sequence such that, $ {u_k}$ are piecewise smooth and $u_k=0 $ on the boundary $ {u_k}$ make the ...
1
vote
2answers
61 views

Confused about Dirichlet L-functions and their poles.

I read : if χ is principal, then the corresponding Dirichlet L-function has a simple pole at $s = 1$. Probably I need to know what principal is and that will solve my question. My confusion was ...
0
votes
1answer
51 views

What are the periodic Dirichlet series?

My question is: What are the periodic Dirichlet series? Does the Riemann zeta function $ζ(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ and the alternating zeta function $η(s)=\sum_{n=1}^\infty (-1)ⁿ⁻¹/n^{s}$ ...
5
votes
2answers
122 views

Looking for explanation of bound on Dirichlet's L-Function

I am reading Stein and Shakarchi's Fourier Analysis text and the proof Dirichlet's theorem and I am looking for clarification on how he derives the following for large $s$, $\lim_{s\to\infty}$ and ...
1
vote
0answers
43 views

A question about the behavior of Dirichlet series and its derivatives

Let us consider the Dirichlet series: $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ We know that its derivative is given by: $$f'(s)=-\sum_{n=1}^\infty \frac{(\ln n) a_n}{n^s} $$ My question is: What ...
17
votes
2answers
760 views

Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the ...
1
vote
0answers
111 views

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
2
votes
1answer
150 views

Functional equation for Hecke $L$-series

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, Theorem II.10.3, we have Let $L(s,\psi)$ be the Hecke $L$-series attached to the Größencharakter $\psi$. Then $L(s,\psi)$ has ...
1
vote
0answers
103 views

Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
4
votes
1answer
99 views

A numeral system built around Dirichlet series, by analogy of how positional numeral systems are built around power series?

For any natural number and chosen base p, the number admits a unique expression of the form $a_np^n + ... + a_2p^2 + a_1p^1 + a_0$, where $a_k < p$ for all k. This property is effectively what ...
5
votes
1answer
126 views

Interesting phenomenon with the $\zeta(3)$ series

I noticed that if one takes certain partial sums of the series for $\zeta(3)$: $$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$ an interesting phenomenon occurs ...
4
votes
0answers
47 views

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 $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$ ...
0
votes
0answers
75 views

Limits as a representation of the Dirichlet function

I read that the Dirichlet function (1 if Rational, 0 else) can be written as: What is the proof of that? Are those limits commutative? Is there any other closed formula for Dirichlet function? (With ...
16
votes
2answers
338 views

How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?

I'm considering the transfer-function $$ t(x) = \log(1 + \exp(x)) $$ and find the beginning of the power series (simply using Pari/GP) as $$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
0
votes
1answer
288 views

Sines and cosines of angles in arithmetic progression

Prove that if $\phi$ is not equal to $2k\pi$ for any integer $k$, then $$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$ ...
1
vote
0answers
62 views

Can we express $x$ as a Dirichlet series?

Can we express $x$ as a Dirichlet series ? Thus $x = a_0 + a_1 2^{-x} + a_2 3^{-x} + a_3 4^{-x} + ...$ where the $a_i$ are real numbers ? I do not know how to get a nondivergent solution.
2
votes
1answer
189 views

Convergence of the Fourier Transform of the Prime $\zeta$ Functions

I think I found a way to write the truncated Prime $\zeta$ function like this: $$ P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} ...