For questions on Dirichlet series.

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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 ...
21
votes
3answers
1k views

On Dirichlet series and critical strips

(I'll keep this one short) Given a Dirichlet series $$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$ where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
17
votes
2answers
391 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 - ...
15
votes
4answers
689 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 ...
13
votes
1answer
738 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 ...
12
votes
2answers
368 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 ...
11
votes
2answers
212 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 ...
10
votes
2answers
498 views

Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$ D(n) = \sum_{k=1}^{n}d(k) , $$ where $$ d(n) = \sum_{k|n}^{n}1. $$ One can observe the following pattern in the values of ...
8
votes
3answers
353 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 ...
8
votes
2answers
659 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 ...
7
votes
3answers
207 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 ...
7
votes
1answer
466 views

Reference request: $L$-series and $\zeta$-functions

Does anyone know a good book, lecture note, article etc. on $L$-series (Dirichlet, Hecke, Artin) and $\zeta$-functions in number theory? I'm especially interested in material explaining the following: ...
6
votes
2answers
118 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 $$ ...
6
votes
2answers
364 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?
5
votes
2answers
848 views

An identity involving the Möbius function

$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$ for $s>1$. How do I prove this identity?
5
votes
1answer
140 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 ...
5
votes
2answers
139 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 ...
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 ...
5
votes
0answers
217 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
4
votes
1answer
131 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 $h(n)\neq 1$?
4
votes
1answer
99 views

Continuous version of the Möbius inversion theorem

Is there a continuous version of Möbius Inversion. Essentially, using integrals instead of sums.
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. ...
4
votes
1answer
74 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 ...
4
votes
1answer
111 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 ...
4
votes
0answers
34 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 ...
4
votes
0answers
52 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 $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$ ...
3
votes
2answers
246 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 ...
3
votes
1answer
110 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 ...
3
votes
3answers
106 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 ...
3
votes
2answers
73 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 ...
3
votes
1answer
84 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 ...
3
votes
1answer
165 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 $f$.
3
votes
1answer
104 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 ...
3
votes
1answer
52 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 ...
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( ...
3
votes
0answers
142 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
124 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) = ...
3
votes
0answers
161 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 ...
2
votes
2answers
148 views

A question about an identity involving Dirichlet characters

Let $\chi$ be a Dirichlet character $\bmod q$. We have $$\sum_{n=0}^{\infty} (-1)^{n-1} \chi(n) n^{-s}=\prod_p ...
2
votes
3answers
434 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.
2
votes
2answers
181 views

Does this Dirichlet series converge to zero?

Consider the periodic Dirichlet series that has this iterative definition: ...
2
votes
2answers
46 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 ...
2
votes
1answer
220 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 ...
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 $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?
2
votes
2answers
114 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. ...
2
votes
1answer
81 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 ...
2
votes
1answer
58 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)^* = ...
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 ...
2
votes
1answer
180 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 ...
2
votes
1answer
49 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, ...