Tagged Questions

33
votes
4answers
935 views

If 2^x and 3^x are integers, must x be as well?

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 $n^x$ …
2
votes
3answers
142 views

Reference for the expected number of prime factors of n larger than n^alpha is -log alpha

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 …
4
votes
7answers
368 views

Infinite sets of primes of density 0

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 …
16
votes
1answer
511 views

Is a “non-analytic” proof of Dirichlet’s theorem on primes known or possible?

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) …
12
votes
2answers
345 views

Heuristic argument for the prime number theorem?

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 …
23
votes
5answers
673 views

Why does the Riemann zeta function have non-trivial zeros?

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 …
4
votes
3answers
235 views

Riemann hypothesis generalization names: extended versus generalized?

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, …
5
votes
1answer
234 views

Primes of the form a^2+1

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 …
5
votes
1answer
127 views

at which rational points does the Hypergeometric function take rational values

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 …
5
votes
3answers
225 views

PNT for general zeta functions, Applications of.

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 …
19
votes
5answers
728 views

Partial sums of multiplicative functions

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 …
5
votes
0answers
122 views

Analytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1

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 $\ …
2
votes
1answer
173 views

Dirichlet L series and integrals

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 …
4
votes
5answers
426 views

Historical question in analytic number theory

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 …
12
votes
0answers
214 views

constants in Gamma factors in functional equation for zeta functions.

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 …

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