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Results tagged with prime-numbers
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user 26522
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.
15
votes
1
answer
898
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Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?
Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be gener …
- 13.8k
12
votes
Why could Mertens not prove the prime number theorem?
Because the scale is too small in Mertens's theorem, and the prime number theorem as well as the Riemann hypothesis are hidden by the $O(1/\log{X})$ notation.
Indeed, the former amounts to strengthen …
- 13.8k
11
votes
About the prime divisors of values of polynomials
Assume without loss of generality that $P$ is irreducible, and denote by $S_P(X)$ the set of primes $p < X$ that divide some value $P(n)$. Let $G$ be the Galois group of $P$ and $n_1 > 0$ the number o …
- 13.8k