James Maynard

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math.NT2026

Exponential sums over primes

James Maynard, Mayank Pandey, Maksym Radziwiłł

Let $α = a / q + ε$ with $(a, q) = 1$ , $q \leq N^{1/2}$ and $∣ ε ∣ \leq 1/ (q N^{1/2})$ , and let $B := max (q, q N ∣ ε ∣)$ . We show that \[ \Bigl|\sum_{n<N}Λ(n)e(nα)\Bigr|\le N^{o(1)}\Bigl(\frac{N}{B^{1/2…

math.NT2024

On the theory of prime producing sieves

Kevin Ford, James Maynard

We develop the foundations of a general framework for producing optimal upper and lower bounds on the sum $\sum_{p} a_{p}$ over primes $p$ , where $(a_{n})_{x /2 < n \leq x}$ is an arbitrary n…

math.NT2024

New large value estimates for Dirichlet polynomials

Larry Guth, James Maynard

We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length $N$ taking values of size close to…

math.NT2024

An almost sharp quantitative version of the Duffin-Schaeffer conjecture

Dimitris Koukoulopoulos, James Maynard, Daodao Yang

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $ψ : N \to [0, 1/2]$ be a function such that the series $\sum…