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Results tagged with prime-numbers
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user 23827
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.
4
votes
Rate of convergence of the prime zeta function P(2)
An answer to Question 2 follows from the lemma below by letting $y\to\infty$.
Lemma. Suppose $x,y$ are real numbers with $12\leqslant x\leqslant y$ and $p$ denotes a prime. Then
$$\sum_{x<p\leqslant y …
- 1,991
6
votes
2
answers
629
views
Rate of convergence of the prime zeta function P(2)
For an application in statistical group theory, we need explicit upper and lower bounds that an expert in number theory (I am not one) may know how to prove.
Question 1: What are "good" bounds $f_1(x …
- 1,991
2
votes
0
answers
106
views
Roots of unity, vanishing sums and derivatives
Fix integers $n\geqslant1$ and $k\geqslant 0$. For an
integer $i$, the $k$-fold derivative of $x^i$ can be denoted by
$i^{\underline{k}}x^{i-k}$ where $i^{\underline{k}}$ means
$i(i-1)\cdots(i-k+1)$ i …
- 1,991
0
votes
primes dividing binomial coefficients
Your first problem has a simple solution.
Suppose $p$ is a prime and $(n!)_p$ is the $p$-part of $n!$. Dirichlet proved $(n!)_p=p^k$ where $k = (n-s_p(n))/(p-1)$ and $s_p(n)$ is the sum of the base-$p …
- 1,991