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Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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Why is it impossible to invert the analytic continuation of a Dirichlet series?

By Mathematica (and the truncated Euler MacLaurin formula) I know that: $$\zeta(s)=\lim_{k\to \infty } \, \left(\sum _{n=1}^k \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\right) \tag{1}$$ when the real part ...
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96 views

Questions related to the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$

This question is related to the following two functions evaluated with the coefficient function $a(n)=\mu(n)\log(n)$. (1) $\quad f(x)=\sum\limits_{n=1}^x a(n)$ (2) $\quad\frac{\zeta'(s)}{\zeta(s)^2}=...
8
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1answer
69 views

On $\sup|\varphi^{-1}(n)|=+\infty$

I am trying to find an elementary proof of the following fact: Given some $N\geq 2$, there are $N$ distinct integers $a_1,\ldots,a_N$ such that $\varphi(a_1)=\ldots=\varphi(a_N)$ with $\varphi$ ...
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1answer
49 views

Are there any minor extensions of Dirichlet's theorem?

For example, can we say that for $k$ $odd$, there are infinitely many primes of the form $a+bk$, for a fixed $a,b$ with $gcd(a,b)=1$? How about for $k$ $odd$, there are infinitely many primes of the ...
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1answer
54 views

Are there infinitely many primes of the form $k\cdot 2^n+1$ for a fixed $n$ and odd $k$

It is clear from Dirichlet's theorem on arithmetic progressions that for a fixed $n$, there are infinitely many primes of the form $k\cdot 2^n+1$ for a fixed $n$ and $k=1,2,3,..$. However, what if we ...
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1answer
31 views

$L(s,\chi)=1$ for non trivial Dirichlet character modulo a polynomial of degree $1$

Let $m(x)\in \Bbb F_q[X]$ be a monic polynomial of degree $1$. Show that for every non-trivial Dirichlet character modulo $m$ we have: $L(s,\chi)=1$. I have seen a theorem that states that if $\...
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23 views

How to extract coefficients from Dirichlet series.

I know, there was a similar question like this: Can the coefficients of a Dirichlet series be recovered? But i can see that in case when given function(series) has only positive real numbers as a ...
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0answers
18 views

Concave Dirichlet series.

There is given function $$f(x)=\sum_{n=1}^{\infty} \frac{a_{n}}{n^{x}}$$ That function converges for non-negative real numbers and is concave on interval $$<0,1>$$ Also $$f(0)=f(1)$$ I am ...
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0answers
50 views

Zeros of special Dirichlet series.

There is given Dirichlet series: $$f:\mathbb{C}\rightarrow \mathbb{R}_{\ge 0}$$ $$f(x+iy)=\eta(x+iy)\eta(x-iy)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{x+iy}}$$ Where $a(n)=\sum_{d|n}(-1)^{d+\frac{n}{d}}d^{...
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1answer
64 views

Finding properties of the sequence.

There is given a sequence $$\{a_{n}\}_{n>0}$$ of real numbers with following conditions: $$\sum_{n=1}^{\infty}a_{n}>0 ;$$ $$\sum_{n=1}^{\infty}a_{n}\log n=0 ;$$ $$\sum_{n=1}^{\infty}a_{n}(\log ...
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19 views

Abscissa of absolute convergence for a particular Dirichlet series

For $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdot\cdot\cdot p_k^{\alpha_k}$ we denote $\alpha(n)=\alpha_1\alpha_2\cdot\cdot\cdot\alpha_k$. Show that $F(s)=\sum_{n\geq 1}\frac{\alpha(n)}{n^s}$ is absolutely ...
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0answers
8 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 ...
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0answers
87 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 ...
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0answers
24 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 ...
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1answer
48 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})...
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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. ...
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0answers
27 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 ...
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86 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) $\quad f(x)=\sum\limits_{n=1}^{x}A143519(n)$ ...
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1answer
25 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 ...
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1answer
55 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}^...
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0answers
22 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 ...
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1answer
36 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
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1answer
50 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
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1answer
119 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\...
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27 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) ...
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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 \...
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1answer
222 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{\...
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3answers
311 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 $A_s$...
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1answer
62 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 ...
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2answers
232 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 ...
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0answers
51 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}} \...
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1answer
43 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 \...
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4answers
225 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 ...
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1answer
150 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 \...
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1answer
24 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 ...
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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 ...
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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 $\sigma > \delta$ ...
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0answers
45 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(...
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1answer
32 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} \...
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1answer
41 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 ...
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1answer
52 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,....
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1answer
31 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)&...
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2answers
94 views

Analytic continuation of the prime zeta series

The prime zeta series is denoted by $ \sum_p \frac{1}{p^s} $, where $ p $ is a prime number. It is absolutely convergent in the half plane right of the abscissa at $ \sigma_a = 1 $. I have seen ...
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1answer
21 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\ $}...
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1answer
25 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) ...
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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
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0answers
218 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 ...
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1answer
116 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
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0answers
29 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
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0answers
130 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) $\quad \frac{\zeta(s-1)}{\zeta(s)}=\sum\limits_{n=1}^\infty\frac{\phi(n)}{n^s}$ (...