# Questions tagged [analytic-number-theory]

Questions on the use of the methods of real/complex analysis in the study of number theory.

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### Density of primes of the form $x^2+my^2$

I was playing with numbers and have the nice conjectures: Let $m$ be a fixed positive integer, and $\pi(N)$ denote the numbers of primes not exceeding $N$ and $\pi_m(N)$ denote the number of prime ...
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### how to prove this extended prime number theorem?

A Generalized Prime Number Theorem? Conjecture Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest ...
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### application of distribution of primes in arithmetic progressions

I try to understand an application of distribution of primes in arithmetic progressions Let $$f(x) = \sum_{p \leq x p\equiv 3 \bmod 10} 1$$ So computing $f(40) = 3$ i.e. the primes: 3, 13, and 23 ...
### On the proof that $\phi(n)/n$ has a limit law
In this question, $\mathbb{N}$ denotes the set of positive integers. Also, $\overline{\mathrm{d}}$, and $\mathrm{d}$ means upper natural density, and natural densitiy respectively. (They are the ...