Prime numbers, diophantine equations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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31
votes
2answers
2k views

What is known about the sum x^{n^2}/n?

It follows from a general theorem of Honda that the formal group with the logarithm $$ x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots $$ has integer coefficients. I became interested in it because its $p$-...
9
votes
3answers
739 views

Bound the error in estimating a relative totient function

Let $n=p_1^{e_1}\cdots p_k^{e_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is $$\Phi(n)=n-\sum_i\frac n{p_i}+\sum_{i \lt j}\frac n{...
26
votes
1answer
6k views

About Goldbach's conjecture

let's consider a composite natural number $n$ greater or equal to $4$. Goldbach's conjecture is equivalent to the following statement: "there is at least one natural number $r$ such as $(n-r)$ and $(n+...
21
votes
2answers
3k views

Distinct numbers in multiplication table

Consider multiplication table for numbers $1,2,\cdots, n$. How many different numbers are there? That is how many different numbers of the form $ij$ with $1 \le i, j \le n$ are there? I'm interested ...
12
votes
3answers
786 views

Hecke equidistribution

For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore $$ a+bi=p^{1/2}e^{i\varphi} $$ where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
76
votes
4answers
9k views

How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
42
votes
3answers
3k views

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

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 $o(n^2)$?
24
votes
4answers
3k views

Which number fields are monogenic? and related questions

A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From Marcus'...
23
votes
5answers
2k views

Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?

The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here: Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...
9
votes
0answers
802 views

Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+...
61
votes
9answers
8k views

Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points. I am interested to know why Mordell and ...
30
votes
4answers
2k views

Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs. If p=1 mod 4 is a prime, we can define the ...
16
votes
6answers
3k views

Erik Westzynthius's cool upper bound argument: update?

Version 2 of this writeup is available, and includes a newer and simple upper bound thanks to MathOverflow 88777 as well as indirect references to future writeups. Details of further work ...
22
votes
2answers
1k views

Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...
15
votes
0answers
652 views

Groups generated by 3 involutions

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
20
votes
2answers
1k views

State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations $$a^5 + b^5 = c^5 + d^5$$ and $$a^6 + b^6 = c^6 + d^6$$ have no known nontrivial solutions, but $$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$ and $$3^...
15
votes
4answers
2k views

Is the Euler product formula always divergent for 0<Re(s)<1?

It is known that the Euler product formula converges for Re(s)>1. (which represents the Riemann zeta function.) My question: Is the Euler product formula always divergent for 0 < Re(s) < 1 ? ...
6
votes
1answer
314 views

Additivity of upper densities with respect to arithmetic progressions of integers

Let $\mathsf{d}^\star$ be the asymptotic upper density, defined on the power set of positive integers $\mathbf{N}^+$, so that $$ \mathsf{d}^\star\colon \mathcal{P}(\mathbf{N}^+) \to\mathbf{R}\colon X\...
5
votes
2answers
722 views

Integral orthogonal group for indefinite ternary quadratic form

I have the indefinite quadratic form $q(x,y,z) = 19 x^2 + 5 y^2 - z^2.$ It's not my fault. I find, on reflection, that I have no idea how to describe the orthogonal group of this over the integers. ...
13
votes
2answers
644 views

Odd-bit primes ratio

Say that a number is an odd-bit number if the count of 1-bits in its binary representation is odd. Define an even-bit number analogously. Thus $541 = 1000011101_2$ is an odd-bit number, and $523 = ...
4
votes
1answer
529 views

Lower bound for a prime gap occurring infinitely often

In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound ...
43
votes
9answers
7k views

Learning Class Field Theory: Local or Global First?

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...
42
votes
9answers
4k views

Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
50
votes
6answers
4k views

How to recognise that the polynomial method might work

A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson. Theorem. Let $(a_1,b_1),\dots,(...
46
votes
14answers
5k views

What is the high-concept explanation on why real numbers are useful in number theory?

The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless ...
42
votes
5answers
4k views

Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?

1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi_1$ sentence? More formally, 2) Is there a $\Pi_1$ sentence which is provably equivalent to RH in PA? Update (July 2010): So we have ...
33
votes
4answers
4k views

Is there an “elementary” proof of the infinitude of completely split primes?

Let K be a Galois extension of the rationals with degree n. The Chebotarev Density Theorem guarantees that the rational primes that split completely in K have density 1/n and thus there are infinitely ...
28
votes
4answers
5k views

Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
43
votes
6answers
4k views

Algebraic Attacks on the Odd Perfect Number Problem

The odd perfect number problem likely needs no introduction. Recent progress (where by recent I mean roughly the last two centuries) seems to have focused on providing restrictions on an odd perfect ...
34
votes
7answers
6k views

Are some numbers more irrational than others?

Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly ...
49
votes
3answers
5k views

What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...
32
votes
4answers
2k views

Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function?

We see that $$\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\prod_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\prod_p \left(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)}\right)=\prod_p\...
37
votes
3answers
2k views

What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
19
votes
7answers
3k views

Asymptotic density of k-almost primes

Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is $$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\...
29
votes
4answers
2k views

Are most cubic plane curves over the rationals elliptic?

%This is a new version of the original question modified in the light of the answers and comments. The word 'most' in the title is ambiguous. The following is one way of making it precise. Question1:...
16
votes
1answer
1k views

On a Conjecture of Schinzel and Sierpinski

Melvyn Nathanson, in his book Elementary Methods in Number Theory (Chapter 8: Prime Numbers) states the following: A conjecture of Schinzel and Sierpinski asserts that every positive rational number ...
33
votes
4answers
2k views

Are There Primes of Every Hamming Weight?

That is, for every integer $n \in \mathbb{Z}_{>0}$ does there exist a prime which is the sum of $n$ distinct powers of $2$? In this case, the Hamming weight of a number is the number of $1$s in ...
11
votes
5answers
2k views

Rational points on a sphere in $\mathbb{R}^d$

Call a point of $\mathbb{R}^d$ rational if all its $d$ coordinates are rational numbers. Q1. Is the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$ dense in rational points, i.e. does $S$ include a ...
35
votes
2answers
2k views

A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
23
votes
5answers
1k views

Are the two meanings of “undecidable” related?

I am usually confused by questions of the type "could such and such a problem be undecidable", because as far as I know there are two distinct possible meanings of "undecidable". I regard the ...
11
votes
4answers
4k views

Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...
20
votes
1answer
488 views

Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$

I know that number fields have been the object of many statistical experiments. Is there some kind of heuristics for the following? Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic ...
14
votes
3answers
4k views

Which integers can be expressed as a sum of three cubes in infinitely many ways?

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$ If $n$ is a cube or twice a cube, identities exist. Elkies suggests no other polynomial identities are known. ...
25
votes
3answers
3k views

Solve in positive integers $n!=m(m+1)$

Is anybody know a solution of this problem? (Sorry, I've missed one summand in the previous post.)
13
votes
3answers
700 views

Reference Request: Unit Fraction, equally spaced denominators not integer

I've been looking at unit fractions, and found a paper by Erdos "Some Properties Of Partial Sums Of The Harmonic Series" that proves a few things, and gives a reference for the following theorem: $$\...
8
votes
1answer
392 views

Isomorphisms between spaces of test functions and sequence spaces

I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
5
votes
3answers
969 views

Simultaneous diophantine approximation

Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor. Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over $\mathbb{Q}...
12
votes
2answers
2k views

Efficient computation of integer representation as sum of three squares

recently I've been studying the problem of integer representation as sum of three squares. Most of the articles that I've found study the function $r_m(n)$ which counts the number of representations ...
15
votes
3answers
1k views

Does this infinite sum provide a new analytic continuation for $\zeta(s)$?

It is well known that the infinite sum: $$\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ only converges for $\Re(s)>1$. The Dirichlet 'alternating' sum: $$\displaystyle \zeta(s) = \...
12
votes
0answers
450 views

Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers

Definition / Question Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $...