**76**

votes

**4**answers

4k 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 $n^x$ is an integer for ...

**72**

votes

**6**answers

9k 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 proof (though a ...

**65**

votes

**6**answers

7k views

### Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...

**57**

votes

**5**answers

22k views

### Consequences of the Riemann hypothesis

I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...

**48**

votes

**4**answers

1k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**44**

votes

**3**answers

6k 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) | n \in \mathbb{Z} ...

**41**

votes

**4**answers

2k views

### If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...

**41**

votes

**2**answers

5k views

### Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients.
Special case: is Möbius nearly Orthogonal to Morse
!
Harold Calvin Marston Morse (24 March ...

**40**

votes

**1**answer

8k views

### Is the Green-Tao theorem true for primes within a given arithmetic progression?

Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...

**40**

votes

**3**answers

3k views

### Every prime number > 19 divides one plus the product of two smaller primes?

This is a part of my answer to this question I think it deserves to be treated separately.
Conjecture Let $A$ be the set of all primes from $2$ to $p>19$. Let $q$ be the next prime after $p$. ...

**37**

votes

**2**answers

1k views

### Infinite exponential representation of real numbers

I was thinking about infinite exponential representation of real numbers (like $2=e^{e^{-e^{-e^{e^{-e^{e^{e^{-e^{-e^{-e^{-e^{-e^{e^{-e^{e^{e^{-e^{e^{\cdot^{\cdot^{\cdot}}}}}}}}}}}}}}}}}}}}}$.
The ...

**36**

votes

**15**answers

4k views

### Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...

**36**

votes

**8**answers

5k views

### Why does the Gamma-function complete the Riemann Zeta function?

Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...

**35**

votes

**6**answers

5k views

### How does one use the Poisson summation formula?

While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. ...

**30**

votes

**7**answers

1k views

### How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...

**30**

votes

**9**answers

10k views

### Approaches to Riemann hypothesis using methods outside number theory [closed]

Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.
The ...

**30**

votes

**1**answer

1k views

### Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation
$$
\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.
$$
Basic theorems about Dirichlet series ...

**30**

votes

**1**answer

2k views

### Is the set of primes “translation-finite”?

The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...

**29**

votes

**3**answers

3k views

### The Hardy Z-function and failure of the Riemann hypothesis

David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...

**29**

votes

**1**answer

955 views

### Prime Number Races in 2 Dimensions

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto
\sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective?
In 1999, when I was an undergraduate student, ...

**28**

votes

**1**answer

5k views

### What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

Hi,
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] ...

**28**

votes

**2**answers

2k views

### Is Li(x) the best possible approximation to the prime-counting function?

The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. ...

**27**

votes

**2**answers

1k views

### Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?

**27**

votes

**2**answers

2k views

### Class Numbers and 163

This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the ...

**25**

votes

**5**answers

2k 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 ...

**25**

votes

**6**answers

4k views

### What does Mellin inversion “really mean”?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...

**23**

votes

**1**answer

1k views

### How good is “almost all” when it comes to the Riemann Hypothesis?

Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part ...

**22**

votes

**4**answers

1k views

### Primes with more ones than zeroes in their Binary expansion

This question is also motivated by the developement around my old MO question about Mobius randomness. It is also motivated by Joe O'Rourke's question on finding primes in sparse sets.
Let $A$ be ...

**22**

votes

**2**answers

1k views

### What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$

Suppose $m$ is a positive integer. A quantity of interest is
$$
H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right)
$$
The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...

**22**

votes

**4**answers

1k views

### Why do zeta functions contain so much information?

Is there some intuitive explanation why Dedekind zeta functions contain so much information about their number field?
For example the residue at the pole s=1 relates several invariants of the number ...

**22**

votes

**2**answers

1k views

### Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?

Consider the equation
$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$
``proved'' by Ramanujan. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = ...

**22**

votes

**4**answers

2k views

### Modular forms and the Riemann Hypothesis

Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function ...

**21**

votes

**3**answers

1k views

### How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set
$\{1^{a_1} \cdot ...

**21**

votes

**3**answers

1k views

### Understanding zeta function regularization

I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced zeta function regularization in a compelling way. The goal is to define the determinant of a positive ...

**21**

votes

**3**answers

1k views

### Discrete Fourier Transform of the Möbius Function

Consider the Möbius function $\mu (m)$. (Thus $\mu(m)=0$ unless all prime factors of $m$ appear once and $\mu (m)=(-1)^r$ if $m$ has $r$ distinct prime factors.) Next consider for some natural number ...

**21**

votes

**2**answers

1k views

### Does the quadratic form $x^2 - 7y^2$ represent infinitely many primes, with the restriction that $0 < y < x/10$?

Surely yes, and in more generality, but can it be proved?
It seems that most, if not all, statements about quadratic forms representing primes fall back on algebraic number theory (i.e. splitting of ...

**21**

votes

**2**answers

1k views

### Is there any sense in which Dirichlet density is “optimal?”

A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| ...

**20**

votes

**9**answers

2k views

### When does the zeta function take on integer values?

Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.
Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann ...

**20**

votes

**2**answers

838 views

### Most squares in the first half-interval

It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...

**20**

votes

**6**answers

3k views

### explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...

**20**

votes

**7**answers

2k views

### What should be learned in an introductory analytic number theory course?

Hello all --
I have the privilege of teaching an introductory graduate course in analytic number theory at the University of South Carolina this fall. What topics should I definitely cover?
I'm not ...

**20**

votes

**1**answer

1k views

### Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
...

**20**

votes

**1**answer

1k views

### Provable zero-free region for any entire function that analytically is similar to zeta(s)

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$:
$f(z)$ is bounded when $\Re z>1+\delta$
$f(z)$ is unbounded when $\Re z=1$
$f(z)$ grows ...

**19**

votes

**4**answers

2k views

### The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following:
Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...

**19**

votes

**4**answers

1k views

### Given a prime $p$ how many primes $\ell<p$ of a given quadratic character mod $p$?

There was this question for which my response was unusally popular, so I dare to ask the following:
(1) Given a prime $p>2$, how many primes $\ell < p$ there exist which are quadratic residues ...

**18**

votes

**4**answers

1k views

### Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "expectional zeros" of course first ...

**18**

votes

**7**answers

3k views

### Complex and Elementary Proofs in Number Theory

The Prime Number Theorem was originally proved using methods in complex analysis. Erdos and Selberg gave an elementary proof of the Prime Number Theorem. Here, "elementary" means no use of complex ...

**18**

votes

**3**answers

914 views

### Finite sums of prime numbers $\geq x$

Let $S_x$ be the set of finite sums of prime numbers $\geq x$. In other words, let $S_x$ be the submonoid of $(\mathbf{Z}_{\geq 0},+)$ generated by the set $\mathcal{P}_{\geq x}$ of prime numbers ...

**18**

votes

**1**answer

542 views

### What motivated Rademacher's contour along the Ford circles?

Apologies if this question isn't suitable for MathOverflow; I posted it on MSE here but it didn't get a response and it felt like it was on the cusp of being suitable for here.
After Ramanujan and ...

**17**

votes

**6**answers

1k views

### Question on consecutive integers with similar prime factorizations

Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le ...