Exponential sums over primes

Abstract: A classical result of Vinogradov shows that, for any $\alpha$ with $$ \Big | \alpha - \frac{a}{q} \Big | \leq \frac{1}{q^2} \ , \ q \leq x^{1/2}, $$ and for any $\varepsilon > 0$, we have, $$ \Big | \sum_{p \leq x} e^{2\pi i \alpha p} \Big | \leq C(\varepsilon) x^{\varepsilon} \cdot \Big ( \frac{x}{\sqrt{q}} + x^{4/5} \Big ). $$ with $C(\varepsilon) > 0$ a constant depending only on $\varepsilon$. This has resisted improvements for the past 80 years, beyond refinements to the $x^{\varepsilon}$ term. The $x / \sqrt{q}$ term cannot be improved without eliminating the existence of a Siegel zero. I'll discuss joint work with James Maynard and Mayank Pandey, in which we reduce the exponent $4/5$ appearing in $x^{4/5}$ to $19/24$, which should have various applications to additive problems related to primes.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)