Riemann hypothesis

There has been a remarkable breakthrough towards the Riemann hypothesis (though still very far from fully resolving this conjecture) by Guth and Maynard making the first substantial improvement to a classical 1940 bound of Ingham regarding the zeroes of the Riemann zeta function (and more generally, controlling the large values of various Dirichlet series): https://arxiv.org/abs/2405.20552 Let 𝑁(σ,𝑇) denote the number of zeroes of the Riemann zeta function with real part at least σ and imaginary part at most 𝑇 in magnitude. The Riemann hypothesis tells us that 𝑁(σ,𝑇) vanishes for any σ>1/2. We of course can't prove this unconditionally. But as the next best thing, we can prove zero density estimates, which are non-trivial upper bounds on 𝑁(σ,𝑇). It turns out that the value σ=3/4 is a key value. In 1940, Ingham obtained the bound \(N(3/4,T) \ll T^{3/5+o(1)}\). Over the next eighty years, the only improvement to this bound has been small refinements to the 𝑜(1) error. This has limited us from doing many things in analytic number theory: for instance, to get a good prime number theorem in almost all short intervals of the form \((x,x+x^\theta)\), we have long been limited to the range \(\theta>1/6\), with the main obstacle being the lack of improvement to the Ingham bound. (1/3)