James Maynard

activity

20122026

most citedNew large value estimates for Dirichlet polynomials

4 citations · 8 across the 7 of their papers we have counts for

collaborators

8 papers

math.NT2026

Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture

Manuel Hauke-Treuer, James Maynard, Andrew Pollington

The Duffin-Schaeffer Conjecture, proven in 2020, gives a zero-full dichotomy for the approximation of irrational numbers with reduced fractions for arbitrary approximation function…

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★ 4 cited

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…

math.NT2014

Large gaps between primes

James Maynard

We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t (1 + o (1)) (lo g x) (lo g lo g x) (lo g lo g lo g lo g x) (lo g lo g lo g x)^{- 2}$ for…

math.NT2012★ 1 cited

On the difference between consecutive primes

J. Maynard

Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. The…

math.NT2012★ 3 cited

On the Brun-Titchmarsh Theorem

J. Maynard

The Brun-Titchmarsh theorem shows that the number of primes $\leq x$ which are congruent to $a (mod q)$ is $\leq (C + o (1)) x / (ϕ (q) lo g x)$ for some value $C$ depending on $\log{x}/\lo…