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Results tagged with prime-numbers
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user 8435
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.
13
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Parity of the Prime Counting Function
I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.
Let:
$\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 …
- 3,699
3
votes
0
answers
153
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The behavior of series involving special subsets of the prime numbers
It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ …
- 3,699
0
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prime zeta function when $0<s<1$
This estimate is probably very crude, but here we have that $\sum_{p \leq x} \frac{1}{p^s} \leq \pi(x)\frac{1}{2^s}$.
- 3,699
1
vote
1
answer
204
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Behavior of a quantity related to Fermat's 4n + 1 Theorem
One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to k …
- 3,699