# How good is “almost all” when it comes to the Riemann Hypothesis?

Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part equal to $1/2$ (i.e, lie on the critical line). Let $$\kappa = \liminf_{T \rightarrow \infty} \frac{N_0(T)}{N(T)}.$$ Conrey showed $\kappa \geq 2/5$, which has the interpretation that at least two fifths of the non-trivial zeroes of $\zeta$ lie on the critical line. Feng recently improved this to $\kappa \geq 0.4128$. In a preprint posted on March 23, 2014, Preobrazhenskii and Preobrazhenskaya claim to show that $\kappa \geq 0.47$ and claim to outline a proof that $\kappa = 1$, which has the interpretation that almost all the non-trivial zeroes of $\zeta$ lie on the critical line The preprint is at http://arxiv.org/abs/1403.5786

I have two questions.

1. How many or what parts of the many known consequences of the Riemann hypothesis are corollaries of the statement $\kappa = 1$?

2. REMOVED

• One answer to (1): the Riemann hypothesis implies that $\big| \#\{$primes${}\le x\} - \int_2^x \frac{dt}{\log t} \big| < 2\sqrt x\log x$ for all $x$. This is known to be false if there's even a single violation of the Riemann hypothesis - then there would be arbitrarily large values of $x$ for which $\big| \#\{$primes${}\le x\} - \int_2^x \frac{dt}{\log t} \big| > x^{1/2+\delta}$ for some fixed $\delta>0$. – Greg Martin Mar 26 '14 at 6:07
• About the appropriateness of (2), there is a consistent policy on MO not to allow general discussions on the correctness of preprints, by fear of endless controversy. Discussions about particular points are Okay, though, like "I don't understand the proof of Lemma 3.14 in such preprint". I share your curiosity about (2) (obviously, when a striking result is announced but no complete proof given, there is a large room for skepticism), but I think you should just leave question (1), which is very interesting. – Joël Mar 26 '14 at 13:48
• To complete Greg's comment (which could be an answer) and Stopple's answer, it is not surprising that the $\kappa=1$ conjectures has no consequences because it is compatible with $N(T)-N_0(T) \sim N(T)/\log \log \log \log T$ for example, meaning that even if "almost all" zeros are on the critical line, still almost as many of them are not. Now if we had a good proof that $\kappa=1$, with a good upper bound for $N(T)-N_0(T)$, we could certainly get some arithmetic application. For example, in the extreme case that $N(T)-N_0(T)=O(1)$ (finitely many zeros off the critical line),... – Joël Mar 26 '14 at 18:04
• I'm sure that we could get some good information we don't already have on the distribution of prime on (relatively, but not too) short intervals $[x,x+y]$, because the contribution of the zeros of the line to the explicit formulae for $\pi(x)$ and $\pi(x+y)$ could be proved to cancel each other to some extent. – Joël Mar 26 '14 at 18:06
• @Joël : regardless of its potential correctness, would it be possible to derive a concrete arithmetic application of the strategy exposed in arxiv.org/abs/1805.07741 ? – Sylvain JULIEN Jun 5 '18 at 18:46

Regarding (1), The American Institute of Mathematics survey of the Riemann Hypothesis refers to $\kappa=1$ as the "100% hypothesis", see