Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [dirichlet-series]

For questions on Dirichlet series.

0
votes
0answers
7 views

Step in the dirichlet estimation using posterior of new x

In some notes I'm reading to show how to estimate a new value given a dirchlet distribution parameters, it says: P(x | D) = Integral of P(x|0, d)P(0|D) d0 then ...
1
vote
0answers
78 views

Abscissa of convergence $\sum_{n=0}^{\infty}\frac{\mu(n)}{n^s}$

I have seen statements like $\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$ is convergent for $\Re(s)>1$, and I have seen proof of it being zero (and therefore convergent) when $s = 1$ but haven’t seen ...
0
votes
0answers
22 views

Riemann hypothesis for $L(s,\chi)$ and $L(s,\chi^\sigma)$

If $\sigma \in \text{Gal}(\mathbb{Q}(\zeta_{\infty})/\mathbb{Q})$ do we know or expect that two Dirichlet L-functions $L(s,\chi)$ and $L(s,\chi^\sigma)$ have more in common, especially in term of ...
1
vote
1answer
43 views

Analytic continuation of a series raised to a power raised to a power?

Background I recently realized I could construct the below formula: $$ \lim_{ x \to 1 }(1-x)(\sum_{r=1}^\infty b_r x^{r^\kappa} ) = (\sum_{\tilde r = 1}^\infty \frac{ b_\tilde r }{\tilde r ^\kappa})...
0
votes
0answers
31 views

Convergence & abscissa relationships between reciprocal Dirichlet series

If $\sum_{n=1}^{\infty} a_nn^{-s}$ and $\sum_{n=1}^{\infty} b_nn^{-s}$ are two Dirichlet series - reciprocal of each other - with abscissa of convergence at $\sigma_1$ and $\sigma_2$ respectively. ...
0
votes
0answers
23 views

For any $t\ge m\ge 3$, we have $~\left|\sum_{n=m}^{t}\frac{\chi(n)\ln n}{n^{s}}\right|\leq\frac{\varphi(k)}{2}\frac{\ln m}{m^{s}}$.

Let $s>1$ be real and for any $t\ge m\ge 3$, we have $$\left|\sum_{n=m}^{t}\frac{\chi(n)\ln n}{n^{s}}\right|\leq\frac{\varphi(k)}{2}\frac{\ln m}{m^{s}},$$ where $\chi$ is a non principal ...
0
votes
0answers
42 views

Questions related to Moebius Transform of Characteristic Function of the Primes

Consider the function defined in (1) below related to the fundamental prime counting function $\pi(x)$. Note that A143519(n) is not multiplicative. (1) $\quad f(x)=\sum\limits_{n=1}^{x}A143519(n)$ ...
1
vote
1answer
23 views

isometry relating $L^2(\mathbb{R}, dx)$ and $L^2([0, \infty), \frac{dx}{x})$

What is the basis for functions on the Hilbert space $L^2([0,\infty), \frac{dx}{x})$. I am studying the Mellin transform and I'm trying to understand the role of the functions $n^{it} = e^{it \, \log ...
0
votes
1answer
52 views

Questions on $f(x)=\sum\limits_{n=1}^{x}a(n)$ with an infinite number of positive integer zeros

This question is related to a class of functions that meet the following conditions. (1) $\quad f(x)=\sum\limits_{n=1}^{x}a(n)$ (2) $\quad f(x)=0$ for an infinite number of values of $x\in\mathbb{Z}^...
0
votes
0answers
17 views

Bounds for a character sum $\sum_{n } \dfrac{\eta(n/N)\chi(n)\sin(2\pi \delta n)}{n}$

Let $\chi$ be a primitive Dirichlet character of large modulus $q > 1$, and let $\delta \in \mathbb{R}$ be fixed. Assume $\eta : \mathbb{R} \to [0,1]$ is smooth compactly supported on $[1,2]$. For ...
1
vote
1answer
26 views

Dirichlet Series Derivative

I tried to obtain the derivative of the Direchlet Series- $\sum_{n=1}^{\infty} \dfrac{f(n)}{n^s}$ Differentiating each of the terns, I obtained- $\sum f'(n)n^{-s}+f(n) n^{-s-1}$ However, I should ...
3
votes
1answer
45 views

On a lemma by Newman relating summability and convergence

On page 73 of his book on Analytic Number Theory, Newman presents the following lemma: Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^\infty \frac{a_n}{n}$ exists and $a_n + \log n$ is ...
2
votes
1answer
110 views

Is it true that $\sum\limits_{n=1}^\infty\frac{\chi_{k,1}(n)}{n^s}=\zeta(s)\sum\limits_{d|k}\mu(d)\,d^{-s}$?

Question 1: Is the relationship illustrated in (1) below true where $\chi_{k,1}(n)$ corresponds to the ordering of Dirichlet characters implemented by Mathematica? (1) $\quad\sum\limits_{n=1}^\infty\...
0
votes
0answers
22 views

Convergence of Dirichlet series via Taylor series

Suppose $f:\mathbb{N}_{\geq 1}\to \mathbb{C}$ is an arithmetic function and $$F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s} $$ the Dirichlet series associated to it. I am trying to prove that if i) F(s) ...
0
votes
0answers
25 views

Bounds for truncated $L$-series over short intervals

Let $\chi$ be a non-principal Dirichlet character. Are there any good non-trivial bounds for short sums of the form $$ \sum_{x < n \leq x + N} \chi(n)n^{i t} $$ as both $x \geq 1$ and $t \in \...
6
votes
1answer
214 views

A rough proof for infinitesimals?

I discovered the following relation for arbitrary $d_r$: $$ \lim_{k \to \infty} \lim_{n \to \infty}\ \sum_{r=1}^n d_r \left( f(\frac{k}{n}r)\frac{k}{n} \right) = \lim_{s \to 1} \! \underbrace{\...
8
votes
3answers
304 views

On the sets of sums $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ with $(a_n)$ periodic and integer valued, for different values of $s$ natural number

For every positive integer $s$, let $A_s$ denote the set of the sums of the converging series $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ for every periodic sequence of integers $(a_n)$. Then each $A_s$...
1
vote
1answer
58 views

Is there a zeta function(with a Dirichlet series) having known roots off the critical line?

Is there a zeta function(with a Dirichlet series) having known roots off the critical line? I thought there was something like the Hilldebrand-Davis zeta function or something like that, but I can't ...
4
votes
2answers
215 views

What is $f(2s+1)$ when $f(s)=\sum_{n=0}^\infty {\frac{(-1)^n}{(2n+1)^s}}=1-\frac{1}{3^s}+\frac{1}{5^s}-\frac{1}{7^s}+\dots$? [duplicate]

Is there an exact form of $$f(s)=\sum_{n=0}^\infty {\frac{(-1)^n}{(2n+1)^s}}=1-\frac{1}{3^s}+\frac{1}{5^s}-\frac{1}{7^s}+\dots$$ when $s$ is odd? Discussion I have been exploring infinite series ...
0
votes
0answers
50 views

Imaginary component of Dirichlet Eta Function's root with real component equal to 1/2

Let $$\space\space\space \eta(z) = \sum_{a=1}^{\infty} \frac{1}{a^{z}} \cdot (-1)^{a-1} $$ Now let $\space$$z = \sigma + it $ , $$\eta(\sigma + it) = \sum_{a=1}^{\infty} \frac{1}{a^{\sigma + it}} \...
1
vote
1answer
38 views

Holomorphic but not a Dirichlet series

I want an example of holomorphic map $f : \Omega \to \Bbb C$ where $\Omega$ is the half-plane $\mathrm{Re}(s) > 1$, such that there is no sequence $(a_n)$ of complex numbers with $$f(s) = \sum_{n \...
3
votes
4answers
171 views

Is this class of series all demonstrably transcendental?

Question: For a vector with integer entries $[a_0, a_1, \dots, a_{k-1}]$ is it true that when $\sum_{n=1}^\infty{\frac{a_{n-1 \mod k}}{n}}$ is not divergent it limits to some transcendental number ...
4
votes
1answer
107 views

How are values of the Dirichlet Beta function derivative derived?

Wolfram Mathworld gives the following values for the beta function derivative. $$\beta'(-1) = \frac{2K}{\pi},\quad \beta'(0) = \ln \left[\frac{\Gamma^{2}(\frac{1}{4})}{2\pi\sqrt{2}} \right],\quad \...
0
votes
1answer
22 views

Singularities of ordinary Dirichlet series

Is there an example of an ordinary Dirichlet series such that (a) the Dirichlet series diverges to infinity at the real point (R > 0) on the line of convergence, and (b) R is not a pole of the ...
0
votes
1answer
38 views

Dirichlet series generated by $\mu(n) / \phi(n)$

Let's define $$ A(s) = \sum_{n=1}^{\infty} \frac{\mu(n)}{\phi(n)}n^{-s},$$ i.e. Dirichlet series generated by $\mu(n)/\phi(n)$. I'm curious whether this Dirichlet series can be represented as other ...
2
votes
0answers
35 views

$(f'/f)(s)$ and $f(s)$

Suppose that $f(s)$ is defined by some Euler product (absolutely convergent for $\sigma > 1$), and we happen to know that its logarithmic derivative $(f'/f)(s)$ is analytic for $\sigma > \delta$ ...
2
votes
0answers
37 views

What are the Dirichlet transforms of $\Lambda(n+1)$ and $\frac{\Lambda(n+1)}{\log(n+1)}$?

This question assumes the following definitions. (1) $\quad\psi(x)=\sum\limits_{n\le x}\Lambda(n)\qquad\text{(second Chebyshev function)}$ (2) $\quad\Pi(x)=\sum\limits_{n\le x}\frac{\Lambda(n)}{\log(...
0
votes
1answer
31 views

Equality between an expression with $\sin x$ and $e^{it}$

I define Dirichlet's kernel in the following way: $$D_N(t) = \sum \limits_{k = -N}^{N} e^{2 \pi i kt}.$$ I managed to show that: $$D_N(t) = \sum \limits_{k = -N}^{N} e^{2 \pi i kt} = e^{-2 \pi i Nt} \...
0
votes
1answer
38 views

Divergence of Dirichlet series and what happens to the series to the left of $ \sigma_c $

Consider a Dirichlet series $ \sum_n \frac{1}{n^s} $. At $ s = \sigma_c = 1 $ this series diverges to $ + \infty $ and it similarly diverges to $ +\infty $ for all $ s = \sigma < \sigma_c $. On ...
1
vote
1answer
46 views

Dirichlet series for 1/ζ(s)

Prove that for Re(s)>1 $$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$$ Where $\mu(n)$ is the Möbius function defined by: $\mu(n)=1, \mbox{if }n=1$ $\mu(n)=(-1)^k, \mbox{if }n=p_1,p_2,....
2
votes
1answer
30 views

Convergence of $L$-series

Having a multiplicative homomorphism $\chi:(\mathbb{Z}/f\mathbb{Z})^\times\to\mathbb{C}^\times,$ why the series (I'm a bit vague here...) $$\sum_{n=1}^\infty\frac{\chi(n)}{n^s}$$ converges for $Re(s)&...
1
vote
0answers
61 views

Analytical continuation of the prime zeta series

The prime zeta series $ \sum_p \frac{1}{p^s} $ where $ p $ is a prime number, is absolutely convergent in half plane right of abscissa at $ \sigma_a = 1 $. I have seen several resources asserting it ...
1
vote
1answer
19 views

Solving a radially symmetric Dirichlet problem

How do I solve this Dirichlet problem? $$\left\{ \begin{array}{l l} u_{xx} + u_{yy} = 1 & \quad \mbox{ on $ x^2 + y^2< a^2\ $,} \\ \quad u(x,y) = 0 & \quad \mbox{ on $ x^2 + y^2 = a^2\ $}...
0
votes
1answer
24 views

Radially symmetric solutions of the Dirichlet problem

How do I find the radially symmetric solutions of this Dirichlet problem? $$\left\{ \begin{array}{l l} u_{tt} + u_{yy} = (x^2+y^2)^2 & \quad \mbox{ on $1< x^2 + y^2< 4\ $,} \\ \quad u(x,y) ...
1
vote
0answers
32 views

On $\sum \frac{\mu(n) e^{i n/m}}{n^s}$

Is anything known about the analytic continuation of \[ \sum \frac{\mu(n) e^{i n/m}}{n^s}, \] where $\mu(n)$ is the Mobius function, into any region containing $s = 0$? If I could know any reference ...
9
votes
0answers
206 views

Odd values for Dirichlet beta function

Hello there I want to find a proof for the generating formula for odd values of Dirichlet beta function given by wikipedia: link I searched MSE and didnt find something similar. My try was to start ...
1
vote
1answer
78 views

Dirichlet Series of Absolute value of Mobius Function equals Ratio of Riemann Zeta

I would like to prove this using Euler products: $$\frac {\zeta(s)}{\zeta(2s)} = \sum_{n=1}^{\infty}\frac {\lvert \mu(n) \rvert}{n^s}$$ I have gotten here, but don't know if this is a correct ...
0
votes
0answers
26 views

Proof of McMahon's factorisatio numerorum result

I recently read that if the number of multiplicative partitions of $n$ is $a_n$, McMahon and Oppenheim observed that its Dirichlet series generating function $f(s)$ has the product representation $$f(...
4
votes
0answers
120 views

What is the explicit formula for $\Phi(x)=\sum\limits_{n=1}^x\phi(n)$?

I ran across the following claimed explicit formula for $\Phi(x)$. (1) $\quad\Phi(x)=\sum\limits_{n=1}^x\phi(n)$ (2) $\quad \frac{\zeta(s-1)}{\zeta(s)}=\sum\limits_{n=1}^\infty\frac{\phi(n)}{n^s}$ (...
6
votes
0answers
156 views

Eisenstein series twisted by a Dirichlet character

On page 17 of https://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf, we see a remark where the author mentioned that If $\chi$ is a non-trivial Dirichlet character ...
4
votes
1answer
95 views

Not-too-slow computation of Euler products / singular series

I'd like to compute, to at least a few digits of accuracy, the constants that arise in Hardy-Littlewood conjecture F / Bateman-Horn conjecture, in particular for just a single quadratic polynomial. ...
0
votes
0answers
51 views

Dirichlet problem in terms of a Fourier sine series

Solve the Dirichlet problem $$\left\{ \begin{array}{l l} u_{t} - 2u - u_{xx} =0 & \quad \mbox{$0<x<1,t>0$}, \\ \quad u(x,0) = \begin{cases} x & \textrm{ if $0\le x\le 1/2$} \\ 1-x &...
1
vote
0answers
29 views

Supremum of a Dirichlet series on a vertical line, in the right half-plane

Consider a Dirichlet series $$\phi(s)=\sum_{n=1}^{\infty}\dfrac{a_n}{n^s},\tag{1}$$ such that $$\sup_{s\in\{w: \mathrm{Re}(w)>0\}}\{|\phi(s)|\}<+\infty\tag{2}$$ and the function $N_{\sigma}:(0,+\...
3
votes
1answer
76 views

Question on Proof of the Equivalence of two Coefficient Functions Related to the Dirichlet Series for $\frac{\zeta(s+1)}{\zeta(s)}$

I derived the relationship illustrated in (1) below which I believe converges for $s>0\lor\Re(s)>\frac{1}{2}$ assuming the Riemann hypothesis. The function $rad(n)$ is the radical or square-free ...
0
votes
0answers
17 views

Blowing up of Dirichlet series on the left of its abscissa

Consider a Dirichlet series $$D(a, s) = \sum_{n \geqslant 1} \frac{a(n)}{n^s}.$$ Assume it converges at $s_0$. Do we know something about the rate of blowing up of $D(a, s)$ when $s$ decreases? More ...
0
votes
1answer
88 views

Question with Dirichlet convolution involving Mobius function and divisor function

So my question is: Use the Dirichlet series to show that $\sum_{k|n}\mu(k)d(\frac{n}{k})$ = 1 for all natural numbers n where d(.) is the divisor function. I've just started learning about the ...
0
votes
1answer
36 views

Multiplicative arithmetic function on the unit disk

Suppose $f$ is a multiplicative arithmetic function that takes values inside the unit disk, and let Re$(s)>1$. We define $F(s) = \sum_{n\ge1}^{}\dfrac{f(n)}{n^s}$. I want to show that $$\text{log } ...
0
votes
1answer
52 views

Is $\sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$ expressible in terms of the zeta function?

I am interested if the alternating Dirichlet Lambda function $$\sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$$ can be expressed in terms of the zeta function or another Dirichlet L-Function. I know that for $...
0
votes
1answer
49 views

Solving for variable inside a sum

So a few calcuations have ultimately led me to this expression $$ \sum_{n=1}^\infty \frac{B_n^2\left( \sinh\left( \sqrt 2\, \pi n \right) - \sqrt 2\,\pi n \right)}{4\pi\sqrt 2\, n} = 1 $$ Is there ...
3
votes
0answers
70 views

Dirichlet series, abscissa of absolute convergence $\neq$ abscissa of uniform convergence

It is well known that Dirichlet series, series of the form $$\sum_{n=1}^{\infty}\dfrac{a_n}{n^s},$$ where $\{a_n\}$ is a complex sequence and s is a complex variable, converge in half planes. The ...