For questions on Dirichlet series.

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17
votes
2answers
345 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 - ...
19
votes
3answers
891 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 ...
8
votes
2answers
636 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 ...
10
votes
2answers
429 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 ...
2
votes
1answer
189 views

Convergence of the Fourier Transform of the Prime $\zeta$ Functions

I think I found a way to write the truncated Prime $\zeta$ function like this: $$ P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} ...
5
votes
2answers
636 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?
15
votes
4answers
640 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 ...
0
votes
1answer
314 views

Sines and cosines of angles in arithmetic progression

Prove that if $\phi$ is not equal to $2k\pi$ for any integer $k$, then $$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$ ...
1
vote
0answers
113 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 ...
0
votes
0answers
17 views

Triangle Mesh as a topological disk [duplicate]

I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My ...