Number Theory

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.

Let $\rm{ord}_p(a)$ be the order of $a$ in $( \mathbb{Z} / p \mathbb{Z} )^*$. In 1927, Artin conjectured that the set of primes $p$ for which an integer $a\neq -1,\square$ is a primitive root (i.e. $\rm{ord}_p(a)=p-1$) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk we will study the behaviour of $\rm{ord}_p(a)$ as $p$ varies over primes, in particular we will show, under GRH, that the set of primes $p$ for which $\rm{ord}_p(a)$ is “$k$ prime factors away” from $p-1$ has a positive asymptotic density among all primes except for particular values of $a$ and $k$. We will interpret being “$k$ prime factors away” in three different ways, namely $k=\omega(\frac{p-1}{\rm{ord}_p(a)})$, $k=\Omega(\frac{p-1} {\rm{ord}_p(a)})$ and $k=\omega(p-1)-\omega(\rm{ord}_p(a))$, and present conditional results analogous to Hooley's in all three cases and for all integer $k$. From this, we will derive conditionally the expectation for these quantities. Furthermore we will provide partial unconditional answers to some of these questions. This is joint work with Leo Goldmakher and Greg Martin.

In 1770 Euler observed that $3^3 + 4^3 + 5^3 = 6^3$ and asked if there was another perfect power that equals the sum of consecutive cubes. This captivated the attention of many important mathematicians, such as Cunningham, Catalan, Genocchi and Lucas. In the last decade, the more general equation $x^k + (x+1)^k + \cdots + (x+d)^k = y^n$ began to be studied. In this talk we will focus on this equation. We will see some known results and one of the most used tools to attack this kind of problems. At the end we will show some new results that appear in arXiv:2404.03457.

In 1966, Tate proposed the Artin–Tate conjectures, which expresses special values of zeta function associated to surfaces over finite fields. Conditional on the Tate conjecture, Milne–Ramachandran formulated and proved similar conjectures for smooth proper schemes over finite fields. The formulation of these conjectures already relied on other unproven conjectures. In this talk, I will discuss an unconditional formulation and proof of these conjectures.

I am going to discuss various results on moments of symmetric square L-functions and some of their applications. I will mainly focus on a recent result of R. Khan and M. Young and our improvement of it. Khan and Young proved a mean Lindelöf estimate for the second moment of Maass form symmetric-square L-functions $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the short interval of length $G >> |t_j|^{(1 + \epsilon)/t^{(2/3)}}$, where $t_j$ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ as long as $|t_j|^{(6/7 + \delta)} << t < (2 - \delta)|t_j|$. We obtain a mean Lindelöf estimate for the same moment in shorter intervals, namely for $G >> |t_j|^{(1 + \epsilon)/t}$. As a corollary, we prove a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the interval $|t_j|^{(2/3 + \delta)} << t << |t_j|^{(6/7 - \delta)}$. This is joint work with Olga Balkanova.