Questions tagged [nt.number-theory]

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

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43
votes
2answers
9k 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: $$\Lambda(n)=...
9
votes
0answers
4k views

Is the conjecture A+B=C following correct?

Is the conjecture on A+B=C following correct ? Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write: $...
103
votes
6answers
15k 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 ...
33
votes
3answers
10k 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 $(n-r)$ ...
13
votes
3answers
1k 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{...
152
votes
2answers
56k views

Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that $\frac{a}{b+c} + \...
41
votes
3answers
18k 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. ...
26
votes
7answers
5k 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)!\...
44
votes
2answers
5k 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 $p$-...
21
votes
3answers
2k 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 ...
12
votes
0answers
1k 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+...
105
votes
7answers
25k views

Zagier's one-sentence proof of a theorem of Fermat

Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares. The proof defines an involution of the set $S= \lbrace (x,y,z) \...
47
votes
4answers
7k 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 ...
39
votes
4answers
6k 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'...
17
votes
3answers
4k views

sum of squares in ring of integers

Lagrange proved that every positive integer is a sum of 4 squares. Are there general results like this for rings of integers of number fields? Is this class field theory? Explicitly, suppose a ...
44
votes
1answer
4k 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 ...
53
votes
3answers
5k 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 $o(n^2)$?
32
votes
3answers
6k views

Distinct numbers in multiplication table

Consider the multiplication table for the numbers $1,2,\dots, n$. 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 ...
11
votes
2answers
1k views

distribution of coprime integers

Let $0 < a < 1$ be fixed, and integer $n$ tends to infinity. It is not hard to show that the number of integers $k$ coprime to $n$ such that $1\leq k\leq an$ asymtotically equals $(a+o(1))\...
511
votes
3answers
50k views

Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}$ such that $f\colon\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
32
votes
8answers
10k views

How does one motivate the analytic continuation of the Riemann zeta function?

I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
47
votes
4answers
3k 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 ...
21
votes
5answers
6k 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. Are the rational points dense on the unit sphere $S :\; x_1^2 +\cdots+ x_d^2 = 1$, i.e. does $S$ ...
44
votes
5answers
4k views

The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$

I was asked the following question by a colleague and was embarrassed not to know the answer. Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, ...
47
votes
1answer
2k views

improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...
20
votes
6answers
4k 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 ...
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 ...
21
votes
1answer
1k views

The abc-conjecture as an inequality for inner-products?

The abc-conjecture is: For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have: $$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
17
votes
2answers
1k views

On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in ...
21
votes
4answers
3k views

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

It is known that the Euler product formula converges for $\Re(s)>1$ (and there it represents the Riemann zeta function). My question: Is the Euler product always divergent for $0 < \Re(s) < ...
25
votes
1answer
1k views

Is the Golomb countable connected space topologically rigid?

The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
5
votes
3answers
1k views

Goldbach conjecture and other problems in additive combinatorics

The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance: $S = T$ is the set ...
288
votes
8answers
135k views

Philosophy behind Mochizuki's work on the ABC conjecture

Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
57
votes
9answers
13k 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 ...
74
votes
5answers
16k views

Inaccessible cardinals and Andrew Wiles's proof

In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals. Here's the article Richard Elwes,...
42
votes
8answers
19k 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 ...
55
votes
5answers
9k 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 two ...
55
votes
1answer
13k 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 $d$. ...
34
votes
3answers
3k views

Is there a ring of integers except for Z, such that every extension of it is ramified?

This is probably common knowledge, alas I have to confess my ignorance. In simpler more abstract language, does $\mathcal{O}_K$ being simply connected (having trivial etale $\pi_1$) imply $\mathcal{O}...
52
votes
6answers
3k views

Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ?

The function $\Gamma(s)$ does not have zeros, but $\Gamma(s)\pm \Gamma(1-s)$ does. Ignoring the real solutions for now and assuming $s \in \mathbb{C}$ then: $\Gamma(s)-\Gamma(1-s)$ yields zeros at: ...
33
votes
5answers
9k 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 ...
54
votes
4answers
3k views

An interesting integral expression for $\pi^n$?

I came on the following multiple integral while renormalizing elliptic multiple zeta values: $$\int_0^1\cdots \int_0^1\int_1^\infty {{1}\over{t_n(t_{n-1}+t_n)\cdots (t_1+\cdots+t_n)}} dt_n\cdots dt_1.$...
35
votes
5answers
4k 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 ...
22
votes
4answers
8k 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 ...
44
votes
4answers
3k 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 ...
27
votes
4answers
4k 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 "exceptional zeros" of course first ...
27
votes
6answers
4k views

Infinitely many primes of the form $2^n+c$ as $n$ varies?

At the time of writing, question 5191 is closed with the accusation of homework. But I don't have a clue about what is going on in that question (other than part 3) [Edit: Anton's comments at 5191 ...
17
votes
0answers
893 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 ...
22
votes
2answers
1k views

A closed form for an integral expressed as a finite series of $\zeta(2k+1)$, $\pi^m$ and a rational?

In this paper the following beautiful integral expression for $\zeta(3)$ is derived: $$\zeta(3)=\frac{1}{7}\,\int_0^{\pi} x\,(\pi-x)\csc(x)\, dx$$ In a comment at the end of this question, I ...
7
votes
3answers
1k 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. ...

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