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Results tagged with prime-numbers
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user 21724
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
2
votes
An estimate of the sum related to primes
By juan's answer above, one only needs to show $\sum_{p \leq e^{\frac{2}{\delta}}} \frac{1}{p^{1-\delta}} \leq \log \frac{1}{\delta} + O(1) $. But by estimating $p^{\delta} = 1 + O( \delta \log p )$, …
- 7,249
7
votes
Accepted
Inquiry on the Chebyshev $\theta$ function
No. Littlewood proved that $\theta(x) > x + c \sqrt{x} \log \log \log x$ holds for infinitely many integers $x$, for some $c > 0$. Cf the answer to this Mathoverflow question, noting that $\theta(x) = …
- 7,249
4
votes
Large prime divisors in small intervals
You can simplify Ramachandra's method by bounding the last sum of p.305 using Brun-Titchmarsh inequality (or Montgomery & Vaughan error-term free version of it) instead of Van der Corput's method + Se …
- 7,249
10
votes
Accepted
Generalization of Mertens' theorem
I detail. It suffices to study the "tail" $P(X) = \prod_{p > X} (1- \frac{1}{p^s})^{-1} $. Using $-\log(1-y) = y + O(y^2)$, we get for real $s>1$
$$ \log P(X) = \sum_{p > X} \frac{1}{p^s} + O \left( …
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2
votes
On a coprime generalization of Cramer's conjecture
I will answer a question slightly different from yours. Let $a(n)$ be the smallest integer $\geq n$ such that $a(n) = b c$ with $(b,c) = 1$ and $b \geq n^r$, $c \geq n^{1-r}$. Then :
$a(n) - n \geq …
- 7,249
5
votes
Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
Two very nice answers have already been given, but I would like to add that Michel Balazard has a book in preparation on this topic (written for undergrads), in which he tries to give a deduction of t …
- 7,249