For questions on Dirichlet series.

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3
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3answers
509 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
57 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
65 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
75 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
441 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
96 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
366 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
37 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
85 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
107 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
124 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 ...
1
vote
1answer
196 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
102 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
38 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
119 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
59 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
140 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
47 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 ...
21
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
202 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
230 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
122 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
114 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
143 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 $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$ ...
0
votes
0answers
85 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 ...
17
votes
2answers
404 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 - ...
-1
votes
2answers
587 views

Sines and cosines of angles in arithmetic progression [closed]

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
64 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
214 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}} ...
3
votes
0answers
81 views

Can that two double series representations of the $\eta$/$\zeta$ function be converted into each other?

By an analysis of the matrix of Eulerian numbers(see pg 8) I came across the representation for the alternating Dirichlet series $\eta$: $$ \eta(s) = 2^{s-1} \sum_{c=0}^\infty \left( ...
2
votes
1answer
100 views

Absolute Convergence of Dirichelet Series

The exact series I must show converges absolutely is: $$\sum_{n=1}^{\infty}{\frac{d(n)^r}{n^s}}$$ for $s > 1$, $r\in \mathbb{N}$ and where $d(n)=\#\text{ of divisors of } n$. I've been able to ...
1
vote
1answer
89 views

Multiplicative Dirichlet Series

How do I go about proving that where $f(k)$ is multiplicative that: $$\sum_{k\geq 1}{\frac{f({k})}{k^{s}}}=\prod_{p}\sum_{k\geq 0}{\frac{f(p^{k})}{p^{ks}}}$$ I first tried to use the fundamental ...
3
votes
0answers
152 views

How write Dirichlet character sums for the terms of the von Mangoldt function?

The way to separately write the terms of the von Mangoldt function $\Lambda$ as Dirichlet character sums seems to be: $$\Lambda (1) = \sum\limits_{n=1}^{\infty } \frac{(e^{\Lambda (1)} \chi ...
3
votes
1answer
172 views

Euler product of Dirichlet Series

For $n$ a positive integer, let $f(n)$ be the squarefree part of $n$. Find the Euler product for $\mathfrak D_{f}(s)$ where $\mathfrak D_{f}(s)$ is the Dirichlet Series of $f$.
2
votes
2answers
181 views

Does this Dirichlet series converge to zero?

Consider the periodic Dirichlet series that has this iterative definition: ...
1
vote
2answers
109 views

Summing Lerch Transcendents

The Lerch transcendent is given by $$ \Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}. $$ While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} ...
15
votes
4answers
697 views

$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions

In a paper about Prime Number Races, I found the following (page 14 and 19): This formula, while widely believed to be correct, has not yet been proved. $$ \frac{\int\limits_2^x{\frac{dt}{\ln ...
5
votes
1answer
200 views

Exponentiation of a Dirichlet series

I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic ...
3
votes
1answer
126 views

Dirichlet series 'shifted' by a polynomial

Let $F(x) \in \mathbb{Z}[x]$ and $$ \xi(s) = \sum^\infty_{n=1}g(n)n^{-s} $$ be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series $$ \xi_F(s) = ...
8
votes
2answers
663 views

Approximation of Products of Truncated Prime $\zeta$ Functions

The problem arose, while I was looking at products of power prime zeta functions $$ P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks}, $$ with $k\in \mathbb{N}$ and $s=it$ with real $t$. By using ...
2
votes
2answers
296 views

Convergence of the series $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$

Could you help me to understand for which $x$ this series converge $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?
12
votes
2answers
370 views

regularity of root spacing of $G(z)=\sum\limits_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$

Define, on $\mathbb{C}$: $$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$ A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs): suggests that the roots of $G(z)$ are ...
6
votes
2answers
368 views

Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
2
votes
1answer
181 views

Dirichlet character over Riemann zeta function

Let $\chi$ be a Dirichlet character mod q and let $$L(s,\chi)=\sum_{n\leq x} \frac{\chi(n)}{n^s}.$$ What is the value of $\displaystyle\lim_{s \rightarrow 1} \frac{L(s,\chi)}{\zeta(s)}$ for principal ...
4
votes
1answer
133 views

Relation between zeta function and Dirichlet L-function

Let $$H(s)=\frac{\zeta(s)}{\phi(q)} \sum_{\chi \mod{q}} L(s,\chi)=\sum_{n=1}^{\infty} \frac{h(n)}{n^s}$$ What is the smallest n (as a function of q) such that $h(n)\neq 1$?
3
votes
0answers
163 views

Dirichlet series represents an analytic function

Let $$T(x)=\sum_{n \leq x} t_n$$ and $T(X)=O(x^a)$ for $a \geq 0$. Now let $$F(s)=\sum_{n=1}^{\infty} \frac{t_n}{n^s}$$ What needs to be checked to prove that this Dirichlet series represents an ...
3
votes
3answers
261 views

Approximate Riemann zeta function

Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$. In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function. My question is: Is there a Functional equation for ...
2
votes
2answers
115 views

Convergence of sum in proof that $\Phi(s) - \frac{1}{s-1}$ extends to $\Re(s) > \frac{1}{2}$

Definitions: $\Phi(s) = \displaystyle\sum_{p} \frac{\log p}{p^s}$ where $p$ denotes a prime number. $\zeta(s) = \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^s}$ denotes the Riemann zeta function. ...