The Quasi-Riemann Hypothesis

A well-known company recently dropped a proof of the ‘quasi-Riemann hypothesis’ as part of their theorem dump. Maksym Radziwiłł of New York University said the implications are “staggering”, continuing: In a sense, the vast majority of consequences of RH already follow from quasi-RH. But what is the quasi-Riemann hypothesis, and what are some of these consequences? Since 1897, we’ve known all the nontrivial zeros of the Riemann zeta function lie in this strip in the complex plane: 0 < Re(z) < 1 The quasi-Riemann hypothesis says they lie in some narrower strip a ≤ Re(z) ≤ 1 – a The Riemann hypothesis claims all these zeros lie exactly on the line Re(z) = 1/2 The closer to this line we can show these zeros are, the more regular the distribution of primes must be! First, remember the number of primes less than n is asymptotically n / ln(n). But a more precise estimate of the number of primes less than n is a function called li(n), which is the integral of 1/ln(x) from 0 to n. The…