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Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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8 views

Bayesian bootstrap : is this correct?

So my problem is as follow : I have a given string of characters, and I would like to quantify the uncertainty linked to the probability of each letter types in the string, based on there observed ...
2
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0answers
42 views

Question about Dirichlet Series Related to Formula for $\frac{1}{e}$

This question is related to the three functions defined in (1) to (3) below where $\coth(z)$ gives the hyperbolic cotangent of $z$. (1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)\quad\text{(Mertens ...
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how to find a solution to Dirichlet problem

I got this Dirichlet problem given with the solution but I don't know how to get there . Find a solution to the following problem $$\begin{cases} u_{tt}=c^{2}u_{xx}+e^{t}\sin(5t),&t>0,x\; \...
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14 views

Cauchy's integral formula and Dirichlet series

There is given Dirichlet series: $$f(x+iy)=\sum_{n=1}^{\infty}\frac{a_{n}(y)}{n^{x}}$$ Where $a_{n}(y)$ are real numbers. I would like to find its zeros in the rectangle: $S=\{(x,y):x\in[a,b], y\in[...
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22 views

Extension of an identity related to a Dirichlet series

Let $F(\sigma+it)=\sum_{n=1}^{\infty}a_n n^{-(\sigma+it)}$ a Dirichlet series that is absolutely convergent in the half-plane $\sigma>\sigma_a$ and let $G(\sigma+it)$ be a function such that $$F(\...
4
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1answer
123 views

A formula to find $ \lim_{s \to 1} \frac{1}{\zeta(s)} (\sum_{x_i=1}^\infty \dots \sum_{x_1 =1}^\infty)\frac{1}{ (\sum_{k=1}^i (x_k)^i)^s}$?

Question Using a conjectured formula of mine I believe the following relation to be true: $$ (\int_0^\infty e^{-x^\lambda} dx)^\lambda = \lim_{s \to 1} \frac{1}{\zeta(s)} (\sum_{x_\lambda=1}^\infty \...
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What is the limit of this Dirichlet series?

Background & Motivation I'm trying to verify/disprove the conjectured formula of the weighted integral of $f(x)$: A rough proof for infinitesimals? $$ \lim_{k \to \infty} \lim_{n \to \infty}\ \...
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0answers
19 views

Squared modulus of Dirichlet eta function.

I have a problem: Find formula for $|\eta(x+iy)|^{2}$, where $\eta(x+iy)=\sum_{n\ge1}\frac{(-1)^{n-1}}{n^{x+iy}}$ I calculated it in two ways and i got a contradiction. First method: $|\eta(x+iy)|...
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1answer
28 views

Help with the derivation of the class number formula

I'm interested in the class number formula derived from the Dedekind zeta function, but I have no idea how to derive it. From what I've read, turning the Dedekind zeta function into a normal Dirichlet ...
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Sequence in Dirichlet series.

There is given function: $$f:\mathbb{R}_{>0}\rightarrow \mathbb{R}_{\ge0}$$ In the form: $$f(x)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{x}}$$ where $a_{n}$ is real number for each natural $n$ Let $f(...
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32 views

Prove the series $\sum n^{-1-it}$ is diverge for all real $t$.

Prove that the series $\sum_{n=1}^\infty n^{-1-it}$ diverges for all real $t$. I have shown in the previous exercise that this series is bounded for nonzero $t$, and when $t=0$, it is famous that the ...
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1answer
22 views

Alternating Bertrand series

It is known that $$\frac{\partial^n\zeta(s)}{\partial s^n}=(-1)^n\sum_{k=1}^\infty{\frac{\log^nk}{k^s}}$$ Can the following alternating version of the sum be expressed in terms of well-known ...
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1answer
24 views

Dirichlet-convolution

Above is the definition i got from my note. I was trying to do these and i get stuck when i complete setting up the definition. I am trying to break down $c(n) = ((e_1 - 2e_2) * u)n$ = $( u * e_1 - ...
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1answer
35 views

Show that $\alpha \int_0^{\pi}X^2dx = \int_0^{\pi}(X')^2dx$ holds for $\alpha > 0$

Consider the equation in the form $X′′ + \alpha X = 0$, with Dirichlet boundary conditions at $x = 0$ and $x = π$. a) Multiply the equation by X and integrate from 0 to π, then integrate the ...
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1answer
22 views

Existence of classical Dirichlet series from Lebesgue integrable inverse Mellin transform

Let $f(s)$ be meromorphic in $\mathbb{C}$. Let the following inverse Mellin transform be Lebesgue integrable for all real positive $x$ at some complex point $s$ with some real $c$: $\frac{1}{2\pi i} \...
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reference request - moments zeta function/dirichlet polynomials

I would like to study some material on the moments of the Riemann Zeta function and Dirichlet polynomials (mean value theorems). I was looking both for some introductory material and for some more ...
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1answer
87 views

Dirichlet sum involving coprimes to p#

This exercise seemed straightforward but I have not managed to do the following proof. Let $p\#$ be the product of primes not exceeding p. Let $c(n)$ be the nth coprime to $p\#$ (mod 2,3,...,p). Let $...
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0answers
38 views

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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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}=...
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1answer
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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
50 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
63 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
36 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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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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19 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
53 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
65 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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25 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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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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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
50 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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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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92 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
27 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
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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
31 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
49 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 ...
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1answer
55 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 ...
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1answer
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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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28 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
27 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
349 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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2answers
314 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
71 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
250 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
57 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
54 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
288 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 ...
5
votes
1answer
202 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
30 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 ...