I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number. If $n^x$ …

Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$. It's my underst …

Sorry if the question is too vague or if the examples I look for are too boringly well-known: my knowledge of analytic number theory is rather primitive...... So, here it goes: su …

It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) …

Here is a bad heuristic argument for the prime number theorem. Let n be a positive integer and assume that PNT holds up to n. Then n itself is prime if and only if for each prime p …

This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal …

This is a "names" question. There are two standard directions of generalization of the Riemann hypothesis: one to L-functions (which is used quite a bit in analytic number theory, …

The fact that the Riemann zeta function $\zeta(s)$ and its brethren have a pole at $s=1$ is responsible for the infinitude of large classes of primes (all primes, primes in arithme …

A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{6}{5};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the …

When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general resul …

It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that |&m …

A sequence $a_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a_{mn} = a_m a_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum_n a_n n^{-s}$, absolutely convergent for $\ …

If $f : t \to e^{-xt}$ with $x \geqslant 1$, and $d_n$ is the number of positive integers that divide $n$, I can show that $$ \lim_{\epsilon \to 0^+} \sum_{n\geqslant 1}\frac{\epsi …

The analytic continuation and functional equation for the Riemann zeta function were proved in Riemann's 1859 memoir "On the number of primes less than a given magnitude." What is …

Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this g …