For questions on Dirichlet series.

learn more… | top users | synonyms

9
votes
3answers
228 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
79 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
180 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
45 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
114 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
67 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
77 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, ...
11
votes
2answers
259 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 ...
7
votes
2answers
131 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 $$ ...
2
votes
1answer
105 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
votes
0answers
79 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
63 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
47 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 ...
1
vote
1answer
90 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
134 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 ...
3
votes
1answer
150 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
50 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
39 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 !
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
294 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
68 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
85 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 ...
13
votes
1answer
828 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
39 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 ...
3
votes
3answers
609 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
36 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
62 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
75 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 ...
2
votes
1answer
95 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
562 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
107 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
381 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
42 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 ...
4
votes
1answer
97 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
123 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
1answer
136 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 ...
2
votes
1answer
241 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 ...
4
votes
1answer
105 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
39 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
140 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
62 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
145 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
48 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 ...
22
votes
2answers
1k 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 ...
7
votes
0answers
213 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
248 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
127 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
119 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 ...
6
votes
1answer
145 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
53 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 $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$ ...