The threshold for making squares is sharp up to a factor of 4/pi

You need to know: Perfect square, limits, notation P for the probability, independent events, selection uniformly at random.

Background: We say that a finite sequence S of positive integers has a square dependence if it has a subset $A\subset S$ such that the product $\prod\limits_{n\in A}n$ of all integers in A is a perfect square. For $x>1$ , let us select integers $a_1,a_2,\dots$ independently uniformly at random from $[1,x]$ , and let T be the smallest integer such that the sequence $a_1,a_2,\dots, a_T$ has a square dependence.

The Euler-Mascheroni constant is the limit $\gamma=\lim\limits_{n\to\infty}\left(-\log n+\sum\limits_{k=1}^n\frac{1}{k}\right) = 0.577...$ . Integer n is called y-smooth if all of its prime factors are at most y. Let $\Psi(x,y)$ be the number of y-smooth integers up to x, $\pi(y)$ be the number of primes up to y, and let $J(x) = \min\limits_{2\leq y \leq x} \frac{\pi(y)x}{\Psi(x, y)}$ .

The Theorem: On 3rd November 2008, Ernie Croot, Andrew Granville, Robin Pemantle, and Prasad Tetali submitted to arxiv a paper in which they proved that for any $\epsilon>0$ , $\lim\limits_{x\to\infty} P\left(\frac{\pi}{4}(e^{-\gamma}-\epsilon)J(x) \leq T \leq (e^{-\gamma}+\epsilon)J(x)\right) = 1$ , where $\gamma$ is the Euler-Mascheroni constant.

Short context: How many random integers between 1 and x we should select such that the product of some selected integers is a perfect square? This question is important for understanding the performance of fastest known factorisation algorithms. In 1994, Pomerance conjectured that the number T of integers needed for this exhibits a sharp threshold, that is, $\lim\limits_{x\to\infty} P\left((1-\epsilon)f(x) \leq T \leq (1+\epsilon)f(x)\right) = 1$ for some function $f(x)$ and any $\epsilon>0$ . The authors of the Theorem further conjectured that this holds with $f(x)=e^{-\gamma}J(x)$ . The Theorem proves the upper bound exactly, and the lower bound up to the constant $\frac{\pi}{4}$ .

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

The inhomogeneous uniform Littlewood’s conjecture is true for almost any pair of real numbers

You need to know: Notation $|n|$ for the absolute value of $n$ , limit inferior $\liminf$ , the notion of (Lebesgue) “almost any”.

Background: For real number $x\in{\mathbb R}$ , let $||x||$ denotes the distance from x to the nearest integer.

The Theorem: On 14th October 2008, Uri Shapira submitted to the Annals of Mathematics a paper in which he proved that for almost any pair of real numbers $\alpha,\beta \in {\mathbb R}$ , and for all $\gamma,\delta \in {\mathbb R}$ , $\liminf\limits_{|n|\to\infty}|n|\,\,||n\alpha-\gamma||\,\,||n\beta-\delta||=0$ .

Short context: In the 1930s Littlewood conjectured that for any two real numbers $\alpha$ and $\beta$ , $\liminf\limits_{n\to\infty} n ||n\alpha|| ||n\beta|| = 0$ . This remains open, but it follows from 1909 Borel theorem that the conjecture is true for (Lebesgue) almost any pair of $\alpha,\beta \in {\mathbb R}$ (see here for a stronger result). This corresponds to the special case $\gamma=\delta=0$ of the Theorem. The statement $\forall \gamma,\delta \in {\mathbb R}, \liminf\limits_{|n|\to\infty}|n|\,\,||n\alpha-\gamma||\,\,||n\beta-\delta||=0$ can therefore be called an inhomogeneous uniform version of the Littlewood’s conjecture. Before 2008, it was not know to hold even for a single pair of $\alpha,\beta$ . The Theorem proves it for almost any pair.

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

The Riemann zeta function can be computed at 1/2+it to within t^(-a) in time t^(4/13+o(1))

You need to know: Complex number, real part $\text{Re}(z)$ of complex number z, function of complex variable, meromorphic function, analytic continuation.

The Theorem: On 30th November 2007, Ghaith Hiary submitted to arxiv a paper in which he proved that, given any constant $\lambda$ , there is an effectively computable constant $C=C(\lambda)$ and absolute constant k, such that for any $t>1$ the value of $\zeta\left(\frac{1}{2}+it\right)$ can be computed to within $\pm t^{-\lambda}$ using at most $C(\log t)^k t^{4/13}$ operations.

Short context: Riemann zeta function $\zeta$ is one of the most studied functions in mathematics, and is central to, for example, understanding the distribution of prime numbers. The values $\zeta(z)$ at line $z=\frac{1}{2}+it$ are particularly important. Before 2007, the fastest algorithm to approximately compute $\zeta\left(\frac{1}{2}+it\right)$ had running time $t^{1/3}=t^{0.333...}$ . The Theorem improves the exponent to $\frac{4}{13}\approx 0.307$ .

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

There exist pairs of primes at distance less than C(log p_n)^(0.5)(log log p_n)^2

You need to know: Prime numbers, logarithm, limit inferior $\liminf$ .

Background: Let $p_n$ denotes the n-th prime number.

The Theorem: On 15th October 2007, Daniel Goldston, János Pintz, and Cem Yíldírím submitted to arxiv a paper in which they proved that $\liminf\limits_{n\to\infty}\frac{p_{n+1}-p_n}{\sqrt{\log p_n}(\log \log p_n)^2}<\infty$ .

Short context: Famous twin prime conjecture states that $p_{n+1}-p_n=2$ for infinitely many n, and is wide open. In an earlier work, Goldston, Pintz, and Yíldírím proved that $\liminf\limits_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0$ , that is, there exist consecutive primes which are closer than $\epsilon\log p_n$ for any $\epsilon>0$ . The Theorem improves $\epsilon\log p_n$ to $\sqrt{\log p_n}(\log \log p_n)^2$ . Using methods developed in these papers, Zhang proved in 2013 that $p_{n+1}-p_n \leq B = 7\cdot 10^7$ infinitely often. This result was then improved to $B=246$ .

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

The Hausdorff dimension of the set of singular pairs is 4/3

You need to know: Euclidean space ${\mathbb R}^2$ , norm $||{\bf x}||=\sqrt{x_1^2+x_2^2}$ of vector ${\bf x}=(x_1, x_2) \in {\mathbb R}^2$ , Hausdorff dimension of a subset of ${\mathbb R}^2$ .

Background: Vector ${\bf x}=(x_1, x_2) \in {\mathbb R}^2$ is called singular pair if for every $\delta > 0$ there exists $T_0$ such that for all $T>T_0$ there exist vector ${\bf p}=(p_1, p_2)$ with integer coordinates and integer $q$ , $0<q<T$ , such that $||q{\bf x}-{\bf p}||<\frac{\delta}{\sqrt{T}}$ .

The Theorem: On 27th September 2007, Yitwah Cheung submitted to the Annals of Mathematics a paper in which they proved that the set of all singular pairs in ${\mathbb R}^2$ has Hausdorff dimension $\frac{4}{3}$ .

Short context: Singular pair is a pair of real numbers $(x_1,x_2)$ that can be simultaneously approximated by rational numbers $\frac{p_1}{q}, \frac{p_2}{q}$ with the same not too large denominator, see here for a progress in a related Littlewood conjecture. All points on any line $a_1x_1+a_2x_2+b=0$ with rational coefficients $a_1, a_2, b$ are known to be singular pairs. The set of points on such lines has Hausdorff dimension $1$ . The Theorem shows that the set of all singular pairs is much large.

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

Upper and lower bounds for the number of squarefree d (0 < d < X) such that the equation x^2-dy^2=-1 is solvable

You need to know: Limit $\lim$ , limit inferior $\liminf$ , limit superior $\limsup$ , infinite product $\prod\limits_{j=1}^{\infty}f(j)$ .

Background: For positive functions $g(X)$ and $h(X)$ we write $g(X)\sim h(X)$ if $\lim\limits_{X\to\infty}\frac{g(X)}{h(X)}=1$ , $g(X)=o(h(X))$ if $\lim\limits_{X\to\infty}\frac{g(X)}{h(X)}=0$ , and $g(X)=\Theta(h(X))$ if $0<\liminf\limits_{X\to\infty}\frac{g(X)}{h(X)} \leq \limsup\limits_{X\to\infty}\frac{g(X)}{h(X)}<\infty$ . An integer d is called squarefree if it is divisible by no perfect square other than 1. For any $X>0$ , let $f(X)$ be the number of positive squarefree integers d less than X such that equation $x^2-dy^2=-1$ has a solution in integers $x,y$ . Also, let us define constants $\alpha=\prod\limits_{j=1}^{\infty}(1+2^{-j})^{-1}=0.419...$ and $\beta=\frac{3}{2\pi}\prod\limits_{p\in{\cal P}_1}\sqrt{1-p^{-2}}$ , where ${\cal P}_1$ denotes the set of all prime numbers p such that $p-1$ is divisible by 4.

The Theorem: On 5th July 2007, Étienne Fouvry and Jürgen Klüners submitted to the Annals of Mathematics a paper in which they proved the existence of positive constants $C_1, C_2$ such that $(C_1-o(1))\left(\frac{X}{\sqrt{\log X}}\right) \leq f(X)\leq (C_2+o(1))\left(\frac{X}{\sqrt{\log X}}\right)$ . In fact, they proved this for $C_1=\alpha \cdot \beta$ and $C_2=\frac{2}{3}\beta$ .

Short context: For integer $d>0$ , fraction $\frac{x}{y}$ approximates $\sqrt{d}$ from below if $x^2-dy^2<0$ , and the approximation is best if $x^2-dy^2=-1$ . This is called the negative Pell equation. For how many squarefree d it is solvable? In 1993, Stevenhagen conjectured that $f(X) \sim (1-\alpha)\beta\left(\frac{X}{\sqrt{\log X}}\right)$ . The Theorem proves that $f(X)=\Theta\left(\frac{X}{\sqrt{\log X}}\right)$ with constants $(0.419...)\beta$ and $(0.666...)\beta$ for the lower and upper bounds, respectively. In a later work, Fouvry and Klüners improved the lower bound to $\frac{5\alpha}{4}\beta=(0.524...)\beta$ . This is close to the conjectured constant $(1-\alpha)\beta\approx 0.58\beta$ .

Links: The original paper is available here. See also Section 10.12 of this book for an accessible description of the Theorem.

Go to the list of all theorems

Moments M_k(T) of the Riemann zeta function are at most CT(log T)^(k^2+e)

You need to know: Complex number, absolute value $|z|$ and real part $\text{Re}(z)$ of complex number z, function of complex variable, meromorphic function, analytic continuation, integration.

Background: For a complex number $z$ with $\text{Re}(z)>1$ , let $\zeta(z)=\sum\limits_{n=1}^\infty \frac{1}{n^z}$ . By analytic continuation, function $\zeta(z)$ can be extended to a meromorphic function on the whole complex plane, and it is called a Riemann zeta function. The Riemann hypothesis states that if $\zeta(z)=0$ then either $z=-2k$ for some integer $k>0$ or $\text{Re}(z)=\frac{1}{2}$ . For $k>0$ , function $M_k(T) = \int_0^T |\zeta(1/2+it)|^{2k} dt$ is called a 2k-th moment of $\zeta$ .

The Theorem: On 4th December 2006, Kannan Soundararajan submitted to the Annals of Mathematics a paper in which he, assuming the Riemann hypothesis, proved that for every $k>0$ and $\epsilon>0$ there is a constant $C=C(k,\epsilon)$ such that inequality $M_k(T) \leq C T(\log T)^{k^2+\epsilon}$ holds for all $T>2$ .

Short context: Riemann zeta function $\zeta$ is one of the most studied functions in mathematics, and the Riemann hypothesis (RH) is one of the most important open problems. One line of study related to $\zeta$ is understanding the growth of moments $M_k(T)$ . It was known that (assuming RH) $c_k T(\log T)^{k^2} \leq M_k(T)$ for some $c_k>0$ and the Theorem provides an almost (up to $\epsilon$ ) matching upper bound. In a later work, Harper (again assuming RH) improved the upper bound to $M_k(T) \leq C_k T(\log T)^{k^2}$ , matching with the lower bound up to the constant factor.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 9.12 of this book for an accessible description of the Theorem.

Go to the list of all theorems

The primes contain arbitrarily long polynomial progressions

You need to know: Prime numbers, polynomials, arithmetic progression.

Background: Let ${\mathbb Z}[m]$ denote the set of polynomials in one variable m with integer coefficients.

The Theorem: On 1st October 2006, Terence Tao and Tamar Ziegler submitted to arxiv a paper in which they proved that given any polynomials $P_1,\dots , P_k\in {\mathbb Z}[m]$ with $P_1(0)=\dots=P_k(0)=0$ , there are infinitely many pairs of positive integers $x$ and $m$ such that numbers $x+P_1(m),\dots , x+P_k(m)$ are simultaneously prime.

Short context: The sequence $x+P_1(m),\dots , x+P_k(m)$ is called a polynomial progression. In an earlier work, Green and Tao proved that primes contains arbitrary long arithmetic progressions, which corresponds to the case $P_j(m)=(j-1)m, \, j=1,\dots,k$ . The Theorem generalises this result to polynomial progressions. Moreover, the authors proved that for any $\epsilon>0$ , there are infinitely many pairs $(x,m)$ as in the Theorem with $1 \leq m \leq x^{\epsilon}$ . In addition, they proved that even any positive proportion of primes contains infinitely many polynomial progressions.

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

Computing the forth moment of of Dirichlet L-functions at the central point for prime moduli with a power saving error

You need to know: Complex number, absolute value $|z|$ and real part $\text{Re}(z)$ of complex number z, function of complex variable, meromorphic function, analytic continuation, big O notation. Also, see this previous theorem description for the concept of primitive Dirichlet character modulo integer $q>0$ .

Background: For a prime $q$ , denote ${\cal P}(q)$ the set of all primitive Dirichlet characters $\chi$ modulo q, and let $\phi^*(q)$ be the number of such characters. For $\chi \in {\cal P}(q)$ , and complex number $z$ with $\text{Re}(z)>1$ , let $L(z,\chi)=\sum\limits_{n=1}^\infty \frac{\chi(n)}{n^z}$ . By analytic continuation, function $L(z,\chi)$ can be extended to a meromorphic function on the whole complex plane, and it is called a Dirichlet L-function.

The Theorem: On 22nd September 2006, Matthew Young submitted to the Annals of Mathematics a paper in which he proved that for any prime $q\neq 2$ and any $\epsilon>0$ , $\frac{1}{\phi^*(q)}\sum\limits_{\chi \in {\cal P}(q)}|L(\frac{1}{2},\chi)|^4 = \sum\limits_{i=0}^4 c_i (\log q)^i + O(q^{-\frac{5}{512}+\epsilon})$ , where $c_i$ are some explicitly computable absolute constants.

Short context: Dirichlet L-functions are generalisations of famous Riemann zeta function (which corresponds to the case $\chi(n)=1$ ), and estimating their moments, especially at point $z=\frac{1}{2}$ (which is called the central point), is an important problem in number theory with many applications. The Theorem computes the fourth moment of Dirichlet L-functions at $z=\frac{1}{2}$ for prime moduli q, with error term decreasing as a power of q. Ealier, similar formula was derived (by Heath-Brown in 1979) only for the Riemann zeta function.

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

Every cubic form over integers in 14 variables has a non-trivial zero

You need to know: Polynomial in n variables.

Background: A cubic form over integers in n variables is the polynomial of the form $P(x_1, x_2, \dots, x_n)=\sum\limits_{i=1}^n \sum\limits_{j=1}^n \sum\limits_{k=1}^n c_{ijk}x_ix_jx_k$ , where $c_{ijk}$ are integer coefficients.

The Theorem: On 8th June 2006, Roger Heath-Brown submitted to Inventiones mathematicae a paper in which he proved that, for every cubic form $P(x_1, x_2, \dots, x_n)$ over integers in $n\geq 14$ variables, there exists integers $x^*_1, x^*_2, \dots, x^*_n$ , not all zero, such that $P(x^*_1, x^*_2, \dots, x^*_n)=0$ .

Short context: The classical 1884 Theorem of Meyer states that any indefinite quadratic from (a quadratic from $P(x_1, x_2, \dots, x_n)=\sum\limits_{i=1}^n \sum\limits_{j=1}^n c_{ij}x_ix_j$ is indefinite if it is less than $0$ for some values of variables and greater than $0$ for others) over integers in $n\geq 5$ variables has a non-trivial zero. All cubic forms are indefinite, and it is conjectured that they must have a non-trivial zero if $n \geq 10$ . In 1963, Davenport proved this for $n\geq 16$ . After more then 40 years with no further progress, the Theorem proves this for $n\geq 14$ .

Links: The original paper is available here.

Go to the list of all theorems