In this case study we'll highlight new world records, going 23% beyond the Riemann Hypothesis. To explain, we start with the (last digit of) prime numbers: \[ {\color{blue}{\bf 2}}, {\color{green}{\bf3}}, {\color{orange}{\bf5}}, {\color{red}{\bf7}}, 1{\bf1}, 1{\color{green}{\bf3}}, 1{\color{red}{\bf7}}, 1{\color{purple}{\bf9}}, 2{\color{green}{\bf3}},\ldots \] After some thought, we may realize that no such last digit may be even (after ${\color{blue}{\bf 2}}$ itself), else the whole number is even; nor may ${\color{orange}{\bf5}}$ appear again. In other words, eventually there are only four options for the last digit: ${\bf1}$, ${\color{green}{\bf3}}$, ${\color{red}{\bf7}}$, or ${\color{purple}{\bf9}}$. Moreover, the Prime Number Theorem (for arithmetic progressions) tells us that each option occurs with roughly equal 25% chance, as we look at primes further out along the number line.
So far, we've been considering the last digit of primes in base 10, but this result holds in any base $b$ we may choose! That is, denoting \begin{align*} \pi(x) = \#\{ {\rm primes } \ p \le x\} \quad {\rm and }\quad \pi(x;b,a) = \#\{ {\rm primes } \ p \le x \ {\rm with \ last \ digit} \ a \ {\rm in \ base} \ b\}, \end{align*} the Prime Number Theorem asserts that \begin{align*} \pi(x;b,a) \ \sim \ \frac{\pi(x)}{\phi(b)}, \qquad\qquad\qquad \color{blue}{(*)} \end{align*} for any fixed integer $b$ and any digit $a$ which is relatively prime with $b$, where $\phi(b)$ is Euler's function. In particular, in base 3 (ternary), there is a 50-50% chance for primes to have last digit 1 or digit 2; in base 101, there's an equal 1% chance for any last digit in 1-100 to occur.
There is analogous story to tell for smooth numbers: we say that an integer is $y$-smooth if all its prime factors are smaller than $y$. For example, the $2$-smooth numbers are simply the powers of two, and $3$-smooth numbers are just products of the form $2^i \cdot 3^j$.
Picking $y$ to be small, smooth numbers are 'opposite' or 'dual' to the primes, in a certain sense, since they have many small prime factors, rather than one large prime factor - and it's often helpful to decompose a given integer into a 'smooth part' and a 'rough part' (the latter containing the large primes).
Surprisingly, many of the techniques used to study prime numbers can also be applied to their smooth cousins. Indeed, denoting \begin{align*} \Psi(x, y; b) = \#\{ \text{$y$-smooth numbers} \ n \le x \text{ coprime to $b$}\} \quad {\rm and }\quad \Psi(x, y; b, a) = \#\{ \text{$y$-smooth numbers} \ n \le x \ {\rm with \ last \ digit} \ a \ {\rm in \ base} \ b\}, \end{align*} we have \begin{align*} \Psi(x, y;b,a) \ \sim \ \frac{\Psi(x, y; b)}{\phi(b)}, \qquad\qquad\qquad \color{blue}{(**)} \end{align*} as $x, y \to \infty$ with $y \le x$, for $a$ and $b$ as before.
A natural (and as in turns out, quite deep) question is: how soon should we expect the asymptotics $\color{blue}{(*)}$ and $\color{blue}{(**)}$ to take effect, in terms of the base $b$? By the Seigel-Walfisz theorem, $\color{blue}{(*)}$ holds provided $b$ grows no faster than a power of $\log x$; and if $b$ is also smaller than around $y^6$, then $\color{blue}{(**)}$ also holds. Moreover, the (Generalized) Riemann Hypothesis asserts that $\color{blue}{(*)}$ holds when the base $b$ is less than roughly $\sqrt{x}$. So starting from a seemingly innocent question about last digits, we arrived at one of the deepest problems in all mathematics! Finding a direct solution would win you \$1 million from the Clay Millennium Prizes.
But until this time, mathematicians have made progress taking a more statistical perspective: The Bombieri-Vinogradov theorem from the 1960s implies that $\color{blue}{(*)}$ holds for over 99% of bases $b$ up to $\sqrt{x}$. In other words, if you sample bases $b$ at random up to $\sqrt{x} = x^{0.5}$ (the Riemann hypothesis threshold), then almost surely $\color{blue}{(*)}$ will hold — and a similar result is true for $\color{blue}{(**)}$. This was one of the great achievements of the 20th century, and for many applications in number theory, the Bombieri-Vinogradov theorem is just as good as assuming the Riemann hypothesis itself!
With such perspective, we may now ask: does $\color{blue}{(*)}$ hold for almost all bases, beyond the $\sqrt{x}$-barrier? In other words, can we go beyond the Riemann hypothesis? This remains a major open question in the field, and the Elliot-Halberstam conjecture predicts this for almost all bases $b$ (almost) up to $x$ itself.
In the 1980s, mathematicians broke the square-root barrier in special cases important for applications. Notably for primes, the celebrated 1986 work of Bombieri-Friedlander-Iwaniec showed $\color{blue}{(*)}$ holds on (a sieve-weighted) average over bases $b$ up to $x^{0.571}$. In other words, this may be interpreted as going 15% beyond the Riemann hypothesis. This record of 15% stood for decades until recently, when Maynard reached 20%, extending the average of $\color{blue}{(*)}$ over bases $b$ up to $x^{0.60}$, using suitable weights.
For smooth numbers, Wolke had already shown in the 1970s that $\color{blue}{(**)}$ holds, in a weak sense, on average over $b$ up to the same barrier of roughly $\sqrt{x}$ (for small $y$ and $a$); in 1993, Granville reached the same threshold in a stronger averaging sense. These results were significantly improved by Fouvry-Tenenbaum in 1996 and Drappeau in 2015, who reached the threshold of $x^{0.60}$ — again, 20% beyond the barrier at $x^{0.5}$. The flexible factorization properties of smooth numbers allow one to drop the sieve weights when averaging over $b$; achieving this for primes is still an open problem when $b > \sqrt{x}$.
These world records at 20% had been regarded as natural and difficult barriers to overcome, even conditionally. Nevertheless, with new world records for primes and smooth numbers, we, Jared Duker Lichtman and Alexandru Pascadi, independently reach 23% beyond the square-root barrier.
Both of these works leverage refined bounds for certain multidimensional exponential sums, which have deep connections to (certain generalizations of) modular forms. In fact, if we assume the Selberg eigenvalue conjecture for these generalized modular forms, we may average over bases $b$ all the way up to $x^{0.625}$ — that is, we may go 25% beyond the Riemann hypothesis!
Key aspects in the new proofs come from applying these exponential sum bounds, with greater care paid to the specific shape of sums that arise from arithmetic progressions; as well as optimizing the exponential sum bounds themselves, with respect to potential counterexamples to Selberg's conjecture, and to the bound's uniformity in the last digit $a$. This latter uniformity opens up the study of the Goldbach representations — that is, writing an (even) integer $N$ as a sum of two primes $N=p_1+p_2$ — for the first time beyond the square-root barrier.
You can read more about the work in this Quanta article and also watch this Numberphile video.
Alexandru Pascadi (pictured left) is a postgraduate student in Oxford Mathematics.
Jared Duker Lichtman (right) was a postgraduate student in Oxford Mathematics and is now NSF Postdoctoral Fellow and an incoming Szegő Assistant Professor at Stanford University.