Consequences of a bound on possible counterexamples to Riemann hypothesis

Question

The Riemann hypothesis has many strong consequences in number theory. The question is: would a bound on the number of zeros of Riemann zeta-function in the critical strip with real part not equal 1/2 have significant consequences? For example, what would the statement that a number of such zeros is finite imply?

Answer 1

Such bounds are generally called zero-density estimates (for the Riemann zeta function or more general $L$-functions), and they have significant consequences. Chapter 10 of Iwaniec-Kowalski's Analytic number theory is devoted to this topic.

A famous example of such a result and application is Huxley's theorem (Inventiones Mathematicae 15 (1972), 164-170): for $x^{7/12+\epsilon}<h<x$ the number of primes in $[x,x+h]$ is asymptotically $h/\log x$. Other fascinating consequences include connections with the Lindelöf Hypothesis (see e.g. the work of Paul Turán).

The statement you propose is stronger than the famous Density Hypothesis (which is also implied by the Lindelöf Hypothesis) for the Riemann zeta function. This hypothesis would imply the above statement with $1/2$ in place of $7/12$.

Answer 2

If the number were finite, one would get a drastic improvement of the error term in the prime number theorem as one direct consequence.

Recall that RH gives, and indeed is equivalent to,
$$ \pi(x)- \operatorname{Li}(x) = O(x^{1/2 + \epsilon}) $$ for every $\epsilon>0$. More generally, one has that if $\zeta(s)$ has no zeros in the complex halfplane $H_{\sigma} =\{z \colon \Re(z)> \sigma\}$ then $$ \pi(x)- \operatorname{Li}(x) = O(x^{\sigma + \epsilon}) $$ for every $\epsilon>0$. Now if the number of zeros were finite it would in particular follow that there is some $\sigma_0<1$ such that $H_{\sigma_0}$ contains no zeros of $\zeta$ (recall that all zeros are known to have real part less than $1$ and if their number is finite take the max of the real parts), and we had $$ \pi(x)- \operatorname{Li}(x) = O(x^{\sigma_0 + \epsilon}) $$ for every $\epsilon >0$.

This would be a lot better than the current error term of the form $O(x \exp(-c (\log x)^{3/5}) )$.