>[!abstract] >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 $N^{3/4}$, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate $N(σ,T) ≤ T^{30(1−σ)/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$ ([[Guth & Maynard, 2024]]). >[!tip] Significance >In 1940, Albert Ingham had provided an upper bound for the number of non-trivial zeros that may lie outside the critical line of $1/2$ (but within the critical strip), by proving that there are at most $y^{3/5+c}$ such zeros with an imaginary part of at most $y$, where $c$ is a constant between 0 and 9, for $0.75 \leq x \leq 1$. This paper by Guth and Maynard reduces that upper bound to $y^{13/25}+c$. In other words, the paper shows that "zeros of the Riemann zeta function become rarer the further away they are from the critical straight line [and] the worse the possible violations of the Riemann conjecture are, the more rarely they would occur" (Blomer, 2024, as cited in [[Bischoff, 2024]]). ^cbb9d3 ## Related - [[Guth, 2024]] - [[Maynard, 2024]]