Any infinite sequence of +1s and -1s has infinite discrepancy

You need to know: Notation ${\mathbb N}$ for the set of natural numbers, notation $\sup$ for the supremum.

Background: Let $f:{\mathbb N}\to\{-1,+1\}$ be an infinite sequence such that each $f(i)$ is either $+1$ or $-1$ . The discrepancy of f is $\sup\limits_{n,d\in{\mathbb N}}\left|\sum\limits_{j=1}^n f(jd)\right|$ .

The Theorem: On 17th September 2015, Terence Tao submitted to arxiv a paper in which he proved that the discrepancy of any $f:{\mathbb N}\to\{-1,+1\}$ is infinite.

Short context: Around 1932, Erdős conjectured that for any infinite sequence $f:{\mathbb N}\to\{-1,+1\}$ of $+1$ s and $-1$ s and any integer C, there exist positive integers n and d such that $\left|\sum\limits_{j=1}^n f(jd)\right|>C$ . The problem asking to prove or disprove this conjecture became known as the Erdős discrepancy problem. It is one of Erdős’s most famous problems, attracted a lot of attention, but, before 2014, was open even for $C=2$ . In 2014, Konev and Lisitsa solved the $C=2$ case with enormous computer-assisted proof whose output takes up 13 gigabytes of data. The Theorem proves the conjecture for all C.

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

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For non-square a, x^4-ay^4+ 4axy^2z-2ax^2z^2+a^2z^4 is prime infinitely often

You need to know: Prime numbers.

Background: An integer a is called perfect square if $a=k^2$ for some integer k.

The Theorem: On 17th July 2015, James Maynard submitted to arxiv a paper in which he proved a theorem which implies, in particular, that for any integer $a$ that is not a perfect square, there are infinitely many prime numbers of the form $x^4-ay^4+4axy^2z-2ax^2z^2+a^2z^4$ with integer $x,y,z$ .

Short context: Let P be a polynomial with integer coefficients in one or more variables. If we substitute integer values instead of variables, will we get infinitely many primes as values of P? This problem is wide open even for simple polynomials like $P(x)=x^2+1$ . Friedlander and Iwaniec in 1998 resolves this question affirmatively for polynomial $x^2+y^4$ , Heath-Brown in 2001 – for $x^3+2y^3$ , see here. Instead of considering polynomials one by one, Maynard developed a method for resolving it for many infinite families of polynomials. The Theorem, as stated above, is just one example of application of this method.

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

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Short and long averages of a bounded multiplicative function are effectively related

You need to know: Notation ${\mathbb N}$ for the set of positive integers, coprime integers.

Background: A function $f:{\mathbb N} \to [-1,1]$ is called multiplicative if $f(1)=1$ and $f(ab)=f(a)f(b)$ holds for all coprime $a,b\in {\mathbb N}$ .

The Theorem: On 19th January 2015, Kaisa Matomäki and Maksym Radziwiłł submitted to arxiv a paper in which they proved the existence of absolute constants $B,C$ (one can take $B=20000$ ) such that for any multiplicative function $f:{\mathbb N} \to [-1,1]$ , for any $2\leq h \leq X$ , and any $\delta>0$ , inequality $\left|\frac{1}{h}\sum\limits_{x\leq n \leq x+h}f(n) - \frac{1}{X}\sum\limits_{X\leq n \leq 2X}f(n) \right|\leq \delta + B\frac{\log\log h}{\log h}$ folds for all but at most $CX\left(\frac{(\log h)^{1/3}}{\delta^2h^{\delta/25}} + \frac{1}{\delta^2(\log X)^{1/50}}\right)$ integers $x\in[X,2X]$ .

Short context: Many functions of central importance in number theory (for example, the Möbius function and the Liouville function) are multiplicative. For many applications, it is important to estimate average value $\frac{1}{h}\sum\limits_{x\leq n \leq x+h}f(n)$ of such function in short intervals of length h. The Theorem states that this is approximately equal to the average $\frac{1}{X}\sum\limits_{X\leq n \leq 2X}f(n)$ over “long” interval $[X,2X]$ , which is much easier to estimate. This has a lot of applications, some of them are derived in the same paper.

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

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The largest gap between consecutive primes below X grows at least as log X log_2 X log_4 X/log_3 X

You need to know: Prime numbers.

Background: Let $p_n$ denotes the n-th prime. For $X>2$ , let $G(X)=\sup\limits_{p_n \leq X}(p_{n+1}-p_n)$ be the largest gap between consecutive primes below X. For $X>16$ , let $F(X)=\frac{\log X \log \log X \log \log \log \log X}{\log \log \log X}$ . With notation $\log_r (X)$ for the r-th fold logarithm, this simplifies to $F(X)=\frac{\log X \log_2 X \log_4 X}{\log_3 X}$ .

The Theorem: On 16th December 2014, Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao submitted to arxiv a paper in which they proved the existence of constant $c>0$ such that $G(X) \geq c F(X)$ for all X.

Short context: How fast does the largest gap $G(X)$ between consecutive primes below X grow with X? It is conjectured that $G(X)$ grows like a constant times $\log^2 X$ , but this is wide open. See here for the best upper bound on $G(x)$ . For the lower bound, the authors proved earlier in 2014 that G(X) grows faster than $\frac{\log X \log_2 X \log_4 X}{(\log_3 X)^2}$ , giving the first substantial improvement since 1938. The Theorem improves this bound by a $\log_3 X$ factor. The authors argue that this bound in the limit of the current techniques, and a radically new approach will be needed to improve beyond this.

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

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The largest gap between consecutive primes below X grows faster than log X log_2 X log_4 X/(log_3 X)^2

You need to know: Prime numbers.

Background: Let $p_n$ denotes the n-th prime. For $X>2$ , let $G(X)=\sup\limits_{p_n \leq X}(p_{n+1}-p_n)$ be the largest gap between consecutive primes below X. For $X>16$ , let $f(X)=\frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2}$ . With notation $\log_r (X)$ for the r-th fold logarithm, this simplifies to $f(X)=\frac{\log X \log_2 X \log_4 X}{(\log_3 X)^2}$ .

The Theorem: On 20th August 2014, Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao submitted to arxiv a paper in which they proved that $\lim\limits_{X\to\infty} \frac{G(X)}{f(X)}=\infty$ .

Short context: How fast does the largest gap $G(X)$ between consecutive primes below X grow with X? It is conjectured that $G(X)$ grows like a constant times $\log^2 X$ , but this is wide open. In 1931, Westzynthius proved that $G(X) \geq c\frac{\log X \log_3 X}{\log_4 X}$ for some constant $c>0$ . This was improved to $G(X) \geq c\frac{\log X \log_2 X}{(\log_3 X)^2}$ by Erdős in 1935, and to $G(X) \geq c f(X)$ by Rankin in 1938. This strange-looking bound, surprisingly, standed for almost 80 years, except of improvements by a constant factor. At last, the Theorem proves that $G(X)$ grows faster than $f(X)$ . The same result was proved independently by Maynard. See here for even better lower bound proved in a later work, and here for an upper bound on $G(x)$ .

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

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If a, b, c, and a+b are non-zero squares, then equation ax^2 + by^2 = c l^2 is partition regular

You need to know: Set ${\mathbb Z}$ of integers, ${\mathbb N}$ of positive integers.

Background: We say that integer n is non-zero perfect square if $n=k^2$ for some integer $k\neq 0$ . We say that equation $p(x, y, \lambda) = 0$ is partition regular if for every partition of ${\mathbb N}$ into finitely many classes, one of the classes contains distinct x, y that satisfy the equation for some $\lambda \in {\mathbb Z}$ . Note that $\lambda$ is a special variable which is allowed to take any integer values.

The Theorem: On 4th March 2014, Nikos Frantzikinakis and Bernard Host submitted to arxiv a paper in which they proved, among other results, the following theorem. Let $p(x, y, z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz$ , where $a, b, c$ are non-zero and $d, e, f$ are arbitrary integers. Suppose that the three integers $e^2-4ac$ , $f^2-4bc$ and $(e+f)^2-4c(a+b+d)$ are all non-zero perfect squares. Then the equation $p(x, y, \lambda)=0$ is partition regular.

Short context: A classical theorem of Furstenberg and Sarközy from 1970-th states that if we partition ${\mathbb N}$ into finitely many classes, we can always find $x,y$ from the same class such that $x-y$ is a (non-zero) perfect square. In other words, equation $x-y=\lambda^2$ is partition regular. In 2006, Khalfalah and Szemerédi proved the same for equation $x+y=\lambda^2$ . The Theorem proves a similar result for a large family of quadratic equations. In particular, it implies that the equation $ax^2 + by^2 = c\lambda^2$ is partition regular whenever integers $a, b, c$ , and $a+b$ are non-zero perfect squares. For example, this holds for the equation $16x^2+9y^2=\lambda^2$ .

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

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Every odd integer greater than 5 is the sum of three primes

You need to know: Even and odd integers, prime numbers.

Background: An integer n is the sum of three primes, if there exist prime numbers $p_1, p_2, p_3$ such that $n=p_1+p_2+p_3$ .

The Theorem: On 30th December 2013, Harald Helfgott submitted to arxiv a paper in which he proved that every odd integer $n>5$ is the sum of three primes.

Short context: In 1742, Goldbach conjectured that every integer $n>5$ is the sum of three primes. Goldbach’s conjecture became one of the best-known unsolved problems in number theory and all of mathematics. In 1937, Vinogradov proved the conjecture for all odd integers greater than some (large) constant $n_0$ . In 1956, Borozdkin proved that one can take $n_0=e^{e^{16038}}$ . In 2002, Ming-Chit improved this to $n_0=e^{3100}$ . The Theorem proves the conjecture for all odd integers. The case of even integers remains open.

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

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For any k, there exist infinitely many bounded intervals containing k primes

You need to know: Prime numbers.

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

The Theorem: On 19th November 2013, James Maynard submitted to arxiv a paper in which he proved that for any positive integer k, there exists a constant $C_k<\infty$ , such that $p_{n+k}-p_n \leq C_k$ for infinitely many values of n. In particular, one can take $C_1=600$ .

Short context: In April 2013, Zhang made a breakthrough by proving the $k=1$ case of the Theorem with constant $C_1=7\cdot 10^7$ . After this, a large group of mathematicians, working online, tried to prove Zhang’s theorem with as small $C_1$ as possible, and, by October 2013, achieved together $C_1=4422$ . On 23rd October, Maynard announced that he, working alone, achieved $C_1=700$ . By the time he submitted the paper to arxiv in November, he improved this to $C_1=600$ . In a later work, group of mathematicians achieved $C_1=246$ . Even more importantly, Maynard’s Theorem works for all k.

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

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The least modulus of a distinct covering system is at most 10^16

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 distinct covering system (or just “covering system” for short) 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 2nd July 2013, Bob Hough submitted to arxiv a paper in which he proved that in every distinct covering system $0<s_1<s_2<\dots<s_k$ we have $s_1 \leq 10^{16}$ .

Short context: A famous problem of Erdős from 1950, known as the minimum modulus problem for covering systems, asks whether for every N there is a distinct covering system S with all $s_i$ greater than N. Despite some partial results, the problem remained open for over 60 years. In 2009, Nielsen constructed a distinct covering system with $s_1=40$ , but conjectured that this cannot be done for arbitrary large N, and the answer to the minimum modulus problem is negative. The Theorem confirms this conjecture.

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

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There exist constant C and infinitely many pairs of primes p,q with 0<q-p<C

You need to know: Prime numbers.

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

The Theorem: On 17th April 2013, Yitang Zhang submitted to the Annals of Mathematics a paper in which he proved the existence of constant $C<\infty$ , such that $p_{n+1}-p_n \leq C$ for infinitely many values of n. The proof implies that one can take $C=7\cdot 10^7$ .

Short context: Famous twin prime conjecture states that $p_{n+1}-p_n=2$ for infinitely many n, and is wide open. Before 2013, it was not even known that $p_{n+1}-p_n\leq C$ infinitely often for any finite constant C. See here and here for best previous upper bounds on $p_{n+1}-p_n$ . The Theorem proves that $p_{n+1}-p_n \leq C$ infinitely often for $C=7\cdot 10^7$ . It initiated work of many researchers to improve the constant C in it. In a reasonably short time, the constant has been improved to $C=600$ and then to $C=246$ .

Links: The original paper is available here.

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