For questions on Dirichlet series.

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3
votes
1answer
117 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) = ...
0
votes
1answer
35 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 ...
5
votes
0answers
211 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
0answers
50 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
0answers
76 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
125 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
0answers
150 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
0answers
40 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...
2
votes
0answers
39 views

Growth rate of arithmetical function

I'm interested in how one would estimate the growth rate of $$f(n)=\sum_{k\le n}\mu^2(k)\log(k)$$ I.e. sum of logarithms of square free integers. I can think of some trivial methods in my head ...
1
vote
0answers
27 views

Abscissa of absolute convergence of a Dirichlet series

I'd like some help to prove the following theorem : Let $\sum_{n \geq 1}\frac{f(n)}{n^s}$ and $\sum_{n \geq 1}\frac{g(n)}{n^s}$ be two Dirichlet series with respective abscissas of absolute ...
1
vote
0answers
19 views

Is $\min \deg$ in a dirichlet subring interesting or is it always $1$?

Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for ...
1
vote
0answers
34 views

Determing sequence from its Dirichlet series

Suppose I know the Dirchlet series $$\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \frac{\zeta(s)}{\zeta(3s)},$$ where $\zeta(s)$ is the usual Riemann zeta function. My question is - is there a way to ...
1
vote
0answers
34 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 ...
1
vote
0answers
44 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 ...
1
vote
0answers
129 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 ...
1
vote
0answers
108 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!} + ...
1
vote
0answers
62 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.
0
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0answers
17 views

Abscissa of convergence for a Dirichlet series

Let $\alpha \in \mathbb{Z}$ and $f(n) = n^{i \alpha n}$. What is the abscissa of convergence, $\sigma_c$, for the associated Dirichlet series, $\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$? Since $|f(n)| = ...
0
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0answers
25 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
0
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0answers
38 views

Dirichlet L series estimation

let $\chi$ be a non-principal character modulo $q$, $M\geq 1$. I have to prove that, if $\vert \sigma - 1 \vert \leq \frac{1}{\log M}$, then $\vert \sum_{n=1}^M \chi (n)n^{-s}\vert\leq 1+e\log M$ and ...
0
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0answers
10 views

Can this matrix be said to consist of Dirichlet characters?

When: $j=1$ the following formula: $$T(n,k)=\prod\limits_{m=1}_{m \mid n}^{\text{n}} \left(\exp ^{-\mu \left(\frac{n}{m}\right)}\left(\Lambda \left(\frac{n}{m}\right)\right) \chi _{\exp ^{-\mu ...
0
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0answers
22 views

Plotting Dirichlet Series $L_{\tau}(s)$ related to Ramanujan tau function in Mathematica

According to wikipedia, Ramanujan tau function $\tau:\mathbb{N}\to\mathbb{Z}$ is defined by equating the coefficients of the power series on both sides of the identity: ...
0
votes
0answers
4 views

The Dirichlet G.F of $2^\Omega$

Let $\Omega(n)$ and $\omega(n)$ be the number of prime factors of $n$ and of distinct prime factors of $n$, respectively. The Dirichlet G.F of $2^\omega$ is well known, and I was wondering if that of ...
0
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0answers
18 views

The inverse of a Dirichlet serie

When we consider a Dirichlet serie : $$F(s)=\sum_{n=1}^{+\infty}{\frac{f(n)}{n^s}}$$ which converges absolutly in the half-plane $\{ \Re(s) > \sigma \}$ with $f(1) \neq 0$, what can we say about ...
0
votes
0answers
29 views

A functional equation for a Dirichlet serie

I'm looking for a functional equation for the following Dirichlet serie $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ? Thank you !
0
votes
0answers
33 views

Convolution Dirichlet series convergence

After theorem 4.1 in http://www.math.uiuc.edu/~hildebr/ant/main4.pdf, which says (paraphrased) Let $f$ and $g$ be arithmetic functions whose Dirichlet series converge absolutely in $s \in ...
0
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0answers
55 views

Generalized Riemann Hypothesis : Zeros of Dirichlet L function and its functional equation

Let $\chi$ pe a primitive character modulo q with $\chi(-1)=1$ ; L is the Dirichlet - L function Define, $\xi(z,\chi)=(q/\pi)^{z/2}\Gamma(z/2)L(z,\chi)$ Show that $L(z,\chi)$ has infinitely many ...
0
votes
0answers
79 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 ...
0
votes
0answers
108 views

$L(1+it,\chi)\neq 0 $ whenever $t \neq 0 \in \mathbb{R}$

I understand that the proof of the assertion in the title uses the same method which proves that zeta function satisfies $\zeta(1+it)\neq 0$, where the above $L$ is Dirichlet L-function. I.e, you ...
-1
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0answers
14 views

Conformal mapping

I know nothing about conformal mapping. I want to find analytical solution of Laplace equation in a hexagon with Dirichlet boundary condition at each wall. I already know the analytical solution of a ...