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Results tagged with analytic-number-theory
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user 7092
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
2
votes
0
answers
53
views
What lower bounds are known on growth of the distribution of the abundancy index?
Let $a(n)=\sigma(n)/n$ be the abundancy index of $n$ and let $F(x)$ be the distribution function of this index: i.e., the proportion of integers $n$ with $a(n)\leq x$. (This function is well-defined a …
- 1,951
8
votes
Accepted
Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $...
Turning my comment into an answer: your conjecture is supported by numerical computation, but much stronger ones are also supported: for instance, Cramér's conjecture, based on models of the primes as …
- 1,951
5
votes
Accepted
Error term for prime harmonic
"Mertens' Proof of Mertens' Theorem" suggests that Mertens had an error term of $O\left(\frac1{\ln x}\right)$, though that's not tight; theorem 14 there offers an $O\left(\frac1{\ln^2x}\right)$ uncond …
- 1,951