Questions tagged [dirichlet-series]
For questions on Dirichlet series.
563
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Proof that ring of formal Dirichlet series is isomorphic to a ring of formal power series over countably many variables
I found this article of E.D. Cashwell and C.J. Everett "The ring of number-theoretic functions" and they said Dirichlet series ring is isomorphic to formal power series ring of countably ...
- 329
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Given a Dirichlet series that diverges, are there conditions to know when the modulus goes off to infinity?
I was working on a problem, and I had made the assumption that given a Dirichlet series
$$
L(s,f)=\sum_{n\geq 1}\frac{f(n)}{n^s}
$$
If I have some $\sigma\in\mathbb{C}$ such that $L(\sigma,f)$ ...
- 2,295
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Dirichlet series for $\frac{\zeta(1-s)}{\zeta(s)}$ [closed]
Wikipedia (here) says that $\frac{\zeta(s-1)}{\zeta(s)}= \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^{s}}$ where $\varphi(n)$ is the totient function. Similarly, is there a known expression involving a ...
- 113
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Convergence of sums in $\ell^p \implies \ell^{p-\epsilon}$
Supose $\displaystyle(b_n)_{n \in \mathbb{N}}$
is a sequence of positive real numbers that
$$\displaystyle\sum_{n \in \mathbb{N}}(b_n)^{2} <\infty.$$
Does exists some $\epsilon>0$ such that $\...
- 33
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What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?
The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by
$$
A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p
$$
is this serie calculated ...
3
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1
answer
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Turán proof that constant sign of Liouville function implies RH
In Mat.-Fys. Medd. XXIV (1948) Paul Turán gives what he says is a proof of the statement that if the summatory $L(x) = \sum_{n\leq x} \lambda(n)$ of the Liouville function $\lambda(n) = (-1)^{\Omega(n)...
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Finding the sum of a series using a Fourier series
I am stuck on how to calculate the value of the following sum:
$\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$
I am aware that you need to find the corresponding function whose Fourier series is represented ...
- 351
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83
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How to compute constants in asymptotic density of numbers divisible by subset of primes
I'm interested in the asymptotic density of the set $S$ of natural numbers divisible only by primes $p \equiv 1 \bmod 4$ (and similar subsets of $\mathbb{N}$). I'm aware of results which show that the ...
- 499
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Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
- 173
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Why does $\sum\limits_{n=1}^\infty \frac{\nu(n)}{n^s} = \sum\limits_{m=1}^\infty \frac{1}{m^s}\sum\limits_p \frac{1}{p^s}$ hold
In context of a exercise about expressing the dirichlet series $$\sum\limits_{n=1}^\infty \frac{\nu(n)}{n^s}$$ in term of the zeta function, where $\nu(n)$ denotes the amount of different prime ...
- 901
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Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.
I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard ...
- 299
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How to interpret a strange formula about $\zeta'(s)/\zeta(s)$
I obtained a strange formula about $\zeta'(s)/\zeta(s)$
$$
\begin{split}
\frac{\zeta'(s)}{\zeta(s)}-(2\pi)^s&\sum_{\Im(\rho)>0} (-i\rho)^{-s}(2\pi)^{-\rho} e^{-i\pi \rho / 2} \Gamma(\rho)\;\;\...
- 78.4k
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Why are these numbers close to $-\log(2)+\text{integer}\,i\pi$?
The following function $f(n)$ has been derived from the Dirichlet eta function:
$$f(n)=\log \left(\sum _{k=1}^n (-1)^{k+1} x^{c \log (k)}\right)-c \log (n) \log (x) \tag{$\ast$}$$
Let: $$s=\rho _1$$ ...
- 7,448
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How to prove the following Dirichlet-series/geometric-series idenity, step by step process?
$$\frac{\zeta(s)}{\zeta(hs)} =\prod_p\left(\frac{1-\frac{1}{p^{hs}}}{1-\frac{1}{p^{s}}}\right) =\prod_p\left(1+\frac{1}{p^s}+\cdots +\frac{1}{p^{(h-1)s}}\right)=\sum_{n\in S_h}\frac{1}{n^s}$$
What is ...
- 103
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Can $\alpha$ be found for $\sum_{n=1}^{\infty}\frac{\sigma_0(n^2)}{\sigma_0(n)}\frac{1}{n^s}=\zeta(s)\sum_{n=1}^{\infty}\frac{\mu^2(n)\alpha }{n^s}$?
I was looking for a pattern among these below:
$$ \sum_{n=1}^{\infty} \frac{\sigma_0(n^2)}{n^s} = \zeta^2(s) \sum_{n=1}^{\infty} \frac{ \mu^2(n)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)} $$
$$ \sum_{n=1}^{...
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