There are (1+o(1))CN^2/log^4 N, C=0.4764… 4-term progressions of primes at most N

You need to know: Prime numbers, arithmetic progression, logarithm, limit, infinite product.

Background: We denote by $o(1)$ any function $f(N)$ such that $\lim\limits_{N\to\infty} f(N)=0$ .

The Theorem: On 4th June 2006, Ben Green and Terence Tao submitted to arxiv a paper in which they proved, among other results, that the number of 4-tuples of primes $p_1<p_2<p_3<p_4 \leq N$ which lie in arithmetic progression is $(1+o(1))C\frac{N^2}{\log^4 N}$ , with $C=\frac{3}{4}\prod\limits_{p\geq 5}\left(1-\frac{3p-1}{(p-1)^3}\right) \approx 0.4764$ , where the product is over all primes $p\geq 5$ .

Short context: In a paper submitted in 2004, Green an Tao proved that, for any k, the set of primes contains infinitely many arithmetic progressions of length k. Moreover, they proved that there are at least $(1+o(1))c_k\frac{N^2}{\log^k N}$ such progressions consisting of primes at most N, where $c_k>0$ are some small constants. The Theorem provides the exact asymptotic count of the 4-term progressions. Moreover, the authors formulated conjectures which imply the exact count of the k-term progressions for all k, and much more. They then proved these conjectures in later works.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.8 of this book for an accessible description of the Theorem.

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The set of all integer solutions of ad-bc=1 is a polynomial family

You need to know: Polynomials in k variables, notation ${\mathbb Z}^n$ for the set of vectors $x=(x_1, x_2, \dots, x_n)$ with integer $x_i$ .

Background: We say that a set $A \subset {\mathbb Z}^n$ is a polynomial family with k parameters if there are exist n polynomials $P_1, P_2, \dots, P_n$ in k variables with integer coefficients such that $x=(x_1, x_2, \dots, x_n)$ belongs to A if and only if there exists integers $y_1, y_2, \dots, y_k$ such that $x_i = P_i(y_1, y_2, \dots, y_k), \, i=1,2,\dots,n$ .

The Theorem: On 16th January 2006, Leonid Vaserstein submitted to the Annals of Mathematics a paper in which he proved that the set of all integer solutions of equation $x_1x_4-x_2x_3=1$ is a polynomial family with 46 parameters.

Short context: What do you mean by “solving” an equation if it has infinitely many solutions? We cannot list all solutions one by one, so the best we can hope for is to present some formulas with parameters which represent all solutions. The Theorem achieves this for the equation $x_1x_4-x_2x_3=1$ , solving a problem which goes back to 1938 paper of Skolem. It immediately implies the existence of polynomial families describing the solutions of some other, more complicated equations.

Links: The original paper is available here. See also Section 10.4 of this book for an accessible description of the Theorem.

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Half of primes have the even sum of digits

You need to know: Prime numbers, relatively prime integers.

Background: Let $q\geq 2$ be an integer. Any integer $n>0$ can be uniquely represented as $n=\sum\limits_{i=1}^{k} b_i q^{k-i}$ , where $k>0$ is an integer and $b_1, b_2, \dots, b_k$ are integers between $0$ and $q-1$ , which are called digits of n in the q-ary number system (if $q=10$ , these are the digits in the usual decimal system). Let $S_q(n)=\sum\limits_{i=1}^k b_i$ be the sum of digits of n. Let $\pi(n)$ be the number of primes less than n, and let $\pi_{q,m,a}(n)$ be the number of primes p less than n such that $S_q(p) - a$ is a multiple of m.

The Theorem: On 10th November 2005, Christian Mauduit and Joël Rivat submitted to the Annals of Mathematics a paper in which they proved that for all integers $m \geq 2$ and $q \geq 2$ such that $q-1$ and $m$ are relatively prime, there exist constants $C_{q,m}>0$ and $\sigma_{q,m}>0$ , such that the inequality $\left|\pi_{q,m,a}(n) - \frac{\pi(n)}{m}\right| \leq C_{q,m} n^{1-\sigma_{q,m}}$ holds for all integers $n>0$ and all integers $a$ .

Short context: Because $C_{q,m} n^{1-\sigma_{q,m}}$ is known to be negligible comparing to $\pi(n)$ for large n, the theorem states that $\pi_{q,m,a}(n) \approx\frac{\pi(n)}{m}$ , that is, the primes whose sum of digits gives any fixed reminder after division by m occupies about $1/m$ of all primes, as intuitively expected. This answers the 1968 question of Gelfond. In the special case $q=10$ and $m=2$ , it states that asymptotically half of all primes have the even sum of digits, and half of primes have the odd sum of digits.

Links: The original paper is available here. See also Section 10.7 of this book for an accessible description of the Theorem.

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A positive answer for Bergelson-Håland question on values of generalized polynomials

You need to know: Polynomial, operation of taking the integer part, limit, complex numbers, complex exponent.

Background: A generalized polynomial is a real-valued function which is obtained from conventional polynomials in one or several variables and applying in arbitrary order the operations of taking the integer part, addition, and multiplication.

The Theorem: On 17th October 2005, Vitaly Bergelson and Alexander Leibman submitted to the Acta Mathematica a paper in which they proved, among other results, that the limit $\lim\limits_{N \to \infty} \frac{1}{N} \sum\limits_{n=1}^N e^{2\pi i u(n)}$ exists for any bounded generalized polynomial $u$ .

Short context: The Theorem gives a positive answer to a 1996 question of Bergelson and Håland. It is just one of the many corollaries of deep theory on the distribution of values of bounded generalized polynomials developed by authors.

Links: The original paper is available here.

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Dyson’s rank partition functions satisfy congruences of Ramanujan type

You need to know: Prime numbers, notation $a\equiv b \,(\text{mod } q)$ if $a-b$ is a multiple of q.

Background: A partition $\lambda$ of an integer $m>0$ is a sequence $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k$ of integers such that $\sum\limits_{i=1}^k \lambda_i = m$ . The difference $r(\lambda)=\lambda_k - k$ is called the rank (or Dyson’s rank) of $\lambda$ . Let $N(r,q; m)$ denote the number of partitions $\lambda$ of m such that $r(\lambda)\equiv r \,(\text{mod } q)$ .

The Theorem: On 10th October 2005, Kathrin Bringmann and Ken Ono submitted to the Annals of Mathematics a paper in which they proved the following result. Let t be a positive odd integer, and let $q$ be a prime such that $6t$ is not divisible by $q$ . If j is a positive integer, then there are infinitely many non-nested arithmetic progressions $An+B$ such that for every $0\leq r < t$ we have $N(r,t; An+B) \equiv 0 \, (\text{mod } q^j) \,\, n = 0,1,2,\dots$ .

Short context: A partition function $p(m)$ is the number of partitions of integer $m>0$ . In 1920, Ramanujan proved that $p(5n+4)$ always divisible by 5, $p(7n+5)$ – by 7, $p(11n+6)$ – by 11. In 2000, Ono proved that for any prime $q \geq 5$ there exist positive integers A and B such that $p(An+B) \equiv 0 \, (\text{mod } q)$ for all $n \geq 0$ (however, there are no prime $q \neq 5, 7, 11$ for which we can take $A=q$ , see here). The Theorem proves that function $N(r,q; m)$ (known as Dyson’s rank partition function) satisfies similar congruences of Ramanujan type.

Links: The original paper is available here. See also Section 10.2 of this book for an accessible description of the Theorem.

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There exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing

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

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

The Theorem: On 10th August 2005, Daniel Goldston, János Pintz, and Cem Yíldírím submitted to arxiv a paper in which they proved that $\liminf\limits_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n} = 0$ .

Short context: Famous twin prime conjecture states that $p_{n+1}-p_n=2$ for infinitely many n, and is wide open. The 1896 prime number theorem implies that the average size of $p_{n+1}-p_n$ is about $\log n$ , hence $\Delta=\liminf\limits_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}$ is at most 1. In 1940, Erdős proved that $\Delta < 1-c$ for some small $c>0$ . This bound was then improved by many authors. In 1988, Maier proved that $\Delta \leq 0.2484...$ , and this remained the best bound until 2005. The Theorem proves that $\Delta=0$ . See here for further progress.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 9.9 of this book for an accessible description of the Theorem.

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The Erdős-Selfridge conjecture for covering systems is true

You need to know: Notation $n\equiv m (\text{mod\,} s)$ if $n-m$ is a multiple of s.

Background: A finite set S of distinct positive integers $0<s_1<s_2<\dots<s_k$ is called a covering system if there exist integers $m_1, m_2, \dots, m_k$ such that each integer n satisfies at least one of the congruences $n\equiv m_i (\text{mod\,} s_i), \, i=1,2,\dots,k$ .

The Theorem: On 25th May 2005, Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, and Gang Yu submitted to the Journal of the AMS a paper in which they proved that for any number B, there is a number $N_B$ , such that inequality $\sum\limits_{i=1}^k\frac{1}{s_k}>B$ holds for any covering system S with $s_1 > N_B$ .

Short context: A famous problem of Erdős from 1950 asks whether for every N there is a covering system S with all $s_i$ greater than N. In 1973, Erdős and Selfridge conjectured that this is not true if we also require that sum of reciprocals $\sum\limits_{i=1}^k\frac{1}{s_k}$ of elements of S is bounded by some constant B. The Theorem confirms this conjecture. In particular, this implies that for any number $K>1$ and N sufficiently large, depending on K, there is no covering system S with all $s_i$ from the interval $(N,KN]$ . In a later work, Hough proved that in fact there is no covering system with $s_1>10^{16}$ , thus solving the original Erdős problem.

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

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The Hasse principle holds for pairs of diagonal cubic forms in s>=13 variables

You need to know: p-adic fields ${\mathbb Q}_p$ .

Background: If a system of equations has a solution with all variables $0$ , we call such solution trivial and all other solutions non-trivial. We say that a collection of systems of equations satisfies the Hasse principle if, whenever one of the systems has a non-trivial solution in real numbers and in all the p-adic fields ${\mathbb Q}_p$ for all primes p, then it has a non-trivial solution in rational numbers.

The Theorem: On 14th April 2005, Jörg Brüdern and Trevor Wooley submitted to the Annals of Mathematics a paper in which they proved that for any integer $s\geq 13$ , and any integer coefficients $a_j, b_j, \, 1\leq j \leq s$ , the system of equations $\sum\limits_{i=1}^sa_ix_i^3 = \sum\limits_{i=1}^sb_ix_i^3 = 0$ has a non-trivial solution in integers if and only if it has a non-trivial solution in the 7-adic field.

Short context: The Hasse principle provides necessary and sufficient conditions for the existence of non-trivial integer solutions in a system of equations. The famous Hasse–Minkowski theorem states that this principle holds for any quadratic equation in s variables in the form $\sum\limits_{i=1}^s \sum\limits_{j=1}^s a_{ij}x_ix_j=0$ with integer coefficients $a_{ij}$ . For cubic equations, the Hasse principle fails in general, but The Theorem implies that it holds for the system $\sum\limits_{i=1}^sa_ix_i^3 = \sum\limits_{i=1}^sb_ix_i^3 = 0$ in $s\geq 13$ variables. This result was earlier known for $s\geq 14$ , and fails for $s\leq 12$ , hence the condition $s\geq 13$ in the Theorem is the best possible.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.9 of this book for an accessible description of the Theorem.

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Pólya-Vinogradov bound is not tight for characters of odd, bounded order

You need to know: Set ${\mathbb Z}$ of integers, greatest common divisor (gcd) of 2 integers, set ${\mathbb C}$ of complex numbers, $o(1)$ notation.

Background: Function $\chi_q:{\mathbb Z}\to{\mathbb C}$ is called a Dirichlet character modulo integer $q>0$ if (i) $\chi_q(n)=\chi_q(n+q)$ for all n, (ii) if $\text{gcd}(n,q) > 1$ then $\chi_q(n)=0$ ; if $\text{gcd}(n,q) = 1$ then $\chi_q(n) \neq 0$ , and (iii) $\chi_q(mn)=\chi_q(m)\chi_q(n)$ for all integers m and n. An example is character $\chi_q^*$ such that $\chi_q^*(n)=1$ whenever $\text{gcd}(n,q) = 1$ and $\chi_q^*(n)=0$ otherwise. This character is called principal and all other characters – nonprincipal. The order of a Dirichlet character $\chi_q$ is the least positive integer m such that $\chi_q(n)^m=\chi_q^*(n)$ for all n. The character $\chi_q$ is called primitive if there is no integer $0<d<q$ such that $\chi_q(a)=\chi_q(b)$ whenever $\text{gcd}(a,q) = \text{gcd}(b,q) = 1$ and $a-b$ is a multiple of d. For any nonprincipal character $\chi_q$ let $M(\chi_q) = \max\limits_{1\leq m\leq q}\left|\sum\limits_{n=1}^m \chi_q(n)\right|$ .

The Theorem: On 2nd March 2005, Andrew Granville and Kannan Soundararajan submitted to the Jornal of the AMS a paper in which they proved that that if $\chi_q$ is a primitive character modulo q of odd order g, then $M(\chi_q) \leq C_g \sqrt{q} (\log q)^{1-\frac{\delta_g}{2} + o(1)}$ where $\delta_g = 1 - \frac{g}{\pi}\sin\frac{\pi}{g}$ , and $C_g$ is a constant depending only on g.

Short context: In 1918, Pólya and Vinogradov independently proved that $M(\chi_q)=O(\sqrt{q}\log q)$ for any nonprincipal Dirichlet character $\chi_q$ . This is known as the Pólya–Vinogradov inequality, has numerous applications, but it is an open question whether a better bound for $M(\chi_q)$ in terms of q is possible. The Theorem provides an improved bound for characters of odd, bounded order.

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

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Linear growth of the number of quintic fields with bounded discriminant

You need to know: Prime numbers, infinite product, field, isomorphic and non-isomorphic fields. Also, see this previous theorem description for the concepts of number field, degree of a number field, and discriminant of a number field.

Background: Number fields of degree $n=5$ are called quintic. For $X>0$ , let $N_n(X)$ denotes the number of non-isomorphic number fields of degree n with absolute value of the discriminant at most X. Also, let ${\cal P}$ be the set of prime numbers.

The Theorem: On 29th September 2004, Manjul Bhargava submitted to the Annals of Mathematics a paper in which he proved that the limit $\lim\limits_{X\to\infty}\frac{N_5(X)}{X}$ exists and is equal to $c_5 = \frac{13}{120}\prod\limits_{p\in {\cal P}}\left(1+p^{-2}-p^{-4}-p^{-5}\right) = 0.149...$ .

Short context: Counting number fields up to isomorphism is a basic and important open problem in the area. There is and old folklore conjecture that $\lim\limits_{n\to\infty}\frac{N_n(X)}{X} = c_n>0$ for every fixed n, but, before 2004, this was known only for $n\leq 3$ (see here for the best upper bounds for $N_n(X)$ available for general n). In June 2004, Bhargava submitted a paper which proves this conjecture for $n=4$ . The Theorem proves it for $n=5$ (quintic fields).

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.11 of this book for an accessible description of the Theorem.

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