All Questions
Tagged with dirichlet-series prime-numbers
22
questions
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$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?
Let $p_n$ be the $n$ th prime number.
Let $f(s)$ be a Dirichlet series defined on the complex plane as :
$$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
- 16.4k
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answer
83
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How to compute constants in asymptotic density of numbers divisible by subset of primes
I'm interested in the asymptotic density of the set $S$ of natural numbers divisible only by primes $p \equiv 1 \bmod 4$ (and similar subsets of $\mathbb{N}$). I'm aware of results which show that the ...
- 499
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Question on convergence of product and Dirichlet series representations of a function
Consider the following two representations of $f(s)$
$$f(s)=
\underset{K\to\infty}{\text{lim}}\left(\prod\limits_{k=1}^K \left(1-\frac{2}{\left.p_k\right.^s}\right)\right)\tag{1}$$
$$f(s)=\underset{N\...
- 7,631
1
vote
1
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For which $s$ does $\sum 1/p^s$ converge?
A well-known result is that $\sum 1/n^s$ converges for $\operatorname{Re}(s)>1$.
Question: For which $s$ does $\sum 1/p^s$ converge, where $p$ is over all primes?
Notes:
Intuitively there are ...
- 3,325
2
votes
1
answer
111
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Product over the primes with relation to the Dirichlet series
What is the value of $\displaystyle \prod_p\left(1+\frac{p^s}{(p^s-1)^2}\right)$
I got this product by defining a function $a(n)$ such that $a(n)=a(p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_n^{a_n})=a_1a_2a_3......
- 73
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1
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How to calculate $ \sum_{n=1}^\infty\frac{3^{\omega(n)}\Omega(n)}{n^s} $?
For $s\in\mathbb C$ with say $\Re s>1$, how to write
$$
\sum_{n=1}^\infty\frac{3^{\omega(n)}\Omega(n)}{n^s}
$$
in terms of the Riemann Zeta function (where $\omega$ is the number of prime factors ...
- 12.1k
3
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143
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Questions about the Dirichlet series $K(s)=\sum\limits_{p^k} p^{\,-k\,s}$
This question is related to Riemann's prime-power counting function $J(x)$, the fundamental prime-counting function $\pi(x)$, and the simple prime-power counting function $k(x)$ defined below where $p\...
- 7,631
2
votes
1
answer
105
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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 $...
- 10.3k
6
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202
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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}=...
- 7,631
2
votes
1
answer
262
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Questions related to Moebius Transform of Characteristic Function of the Primes
Consider the function $f(x)$ defined in (2) below related to the fundamental prime counting function $\pi(x)$ defined in (1) below. Note $b(n)$ is the Möbius transform of $a(n)$.
(1) $\quad \pi(x)=\...
- 7,631
2
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92
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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(...
- 7,631
1
vote
1
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353
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Dirichlet density of $p \equiv 1 \bmod 8$
I am pretty new to this website although it looks really useful. Could someone help me please? My question is that same as this one: Computing the Dirichlet Density
I can't work out how to calculate ...
- 533
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483
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Computing the Dirichlet Density
I have been asked the following question. I'm not even sure where to start.
Compute the Dirichlet density of the set of all primes $p$ such that $2$, $-2$, and $-1$ are all squares modulo $p$.
- 703
1
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164
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Asymptotic for primitive Dirichlet characters over primes
The Polya-Vinogradov's inequality states that if $\chi$ is a primitive character mod $q$ then $|\sum_{n\le x}\chi(n)|\le \sqrt{q}\log q$. I now want to give the estimation for the sum over primes, i.e....
- 173
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77
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On Dirichlet series and Firoozbakht's conjecture
On assumption of the Firoozbakht's conjecture (this is the Wikipedia, but the reference is for Carlos Rivera's Page) one has that can writes informally the Dirichlet series in LHS of this inequality $$...
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