All Questions
Tagged with prime-number-theorem reference-request
11 questions
5
votes
1
answer
736
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Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
user344548
4
votes
1
answer
601
views
Reference for a proof of Euclid's Theorem for the infinitude of primes
I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of ...
- 17.9k
11
votes
4
answers
707
views
Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
- 20.2k
20
votes
2
answers
4k
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information-theoretic derivation of the prime number theorem
Motivation:
While going through a couple interesting papers on the Physics of the Riemann Hypothesis [1] and the Minimum Description Length Principle [2], a derivation(not a proof) of the Prime Number ...
- 3,871
5
votes
0
answers
193
views
Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
0
votes
0
answers
114
views
The best error term for the second moment
Let $r_2(n)$ be the number of representations of a positive integer $n$ as a sum of two prime squares, i.e. $n=p^2+q^2$. Consider $S_1(x)= \sum_{n \le x} r_2(n)$ and $S_2(x) = \sum_{n \le x}r_2^2(n)$. ...
6
votes
0
answers
333
views
Explicit bounds for the Mertens function
It is a consequence of some forms of the prime number theorem that with $\mu$ the Möbius function, for all $A > 0$, there exists $c_A$ such that for all sufficiently large $x$, $$\frac{1}{x}\sum_{n\...
- 1,890
3
votes
1
answer
224
views
PNT analog for primes inside a structured set
Let $\Bbb T$ be the set of all square free integers with ordering derived from $\Bbb N$. Essentially $PNT$ says if you pick $\log N$ integers less than $N$ you can expect one of them to be prime.
...
- 13.9k
1
vote
0
answers
176
views
R.H. equivalent statement condition
Is the inequality $\prod \limits_{p \leq \sqrt{x}} (1+\frac{1}{p^2-1}) \prod \limits_{p \leq x} (1+\frac{1}{p}) \leq e^\gamma \ln(\theta(\sqrt{x})+\theta(x))$ where $\theta(x)$ is the Chebyshev's ...
user95470
10
votes
3
answers
1k
views
Quantitative and elementary proofs of the Prime Number Theorem
I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...
- 1,687
0
votes
1
answer
1k
views
Name of a conjecture on difference of prime numbers? [closed]
Hello Dear
there is a conjecture for which I do not know how it is called. The conjecture is:
Every even number can be always written as the difference between two prime numbers.
Could you please ...
- 3