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What will happen if a mathmatician have prove that 99.999....% of the solution stay on the critical line and receive the prize but after that another mathmatician find finite numbers of interesting solution out of the line?

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I would imagine the mathematician who found the counter examples would demand that the mathematician who received the prize retract his paper and hand over the money. :) –  Bill Cook Aug 29 '12 at 22:53
99.999...%=100% (if ... means repeating forever) –  Graphth Aug 29 '12 at 23:07
@dotdot - 4 over infinity equal to 0... –  Victor Aug 29 '12 at 23:15
A proof that "99.999%" of the zeros are on the critical line, or any similar statement about "most" or "almost all" of the zeros, would not, strictly speaking, constitute a proof of the Riemann hypothesis, and I presume it would not be eligible for the Clay prize. –  Nate Eldredge Aug 31 '12 at 0:57
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closed as not a real question by fpqc, Old John, Asaf Karagila, William, Argon Aug 30 '12 at 1:55

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up vote 6 down vote accepted

This is entirely possible.

In theory, one could prove that 100% of the zeros on the line, that is a set of density $1$, which does not neccesarily imply the Riemann Hypothesis. There still could exist a very sparse set of zeros off the line, a set of density $0$. To be precise, let $$N(T)=\left|\left\{s=\sigma+i\gamma:\ \zeta(s)=0,\ 0\leq \sigma\leq 1\ \text{and} |\gamma|\leq T \right\}\right|$$ be the number of zeros up to height $T$, and let $$N_0(T)=\left|\left\{s=\frac{1}{2}+i\gamma:\ \zeta(s)=0,\ \text{and} |\gamma|\leq T \right\}\right|$$ be the number of zeros on the critical line up to height $T$. Then, proving that $$\lim_{T\rightarrow \infty}\frac{N_0(T)}{N(T)}=1$$ means that almost all of the zeros lie on the critical line, i.e. that the density is $1$, but it does not imply the Riemann hypothesis.

See this Math Overflow question: If the Riemann Hypothesis fails must it fail infinitely-often? There are a lot of good answers on that thread that answer this question precisely.

In this Math Overflow question, Current Status of the Riemann Hypothesis, (thanks Alex Becker for the link) they discuss the existing results in this direction. In 1942 Selberg proved that a positive proportion of the zeros lie on the critical line. In 1974 Levinson showed that this proportion is at least 1/3, and Conrey improved this to 2/5. Just last year, in 2011, Bui, Conrey and Young improved this to 41%

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@GerryMyerson: You are right, I meant density but wrote measure. What I was thinking was extremely clear, but I wrote something slightly different which made no sense. It is now updated. –  Eric Naslund Aug 30 '12 at 4:27
I think it has been proven by Hardy that there are an infinite number of zeros on the critical line, but that doesn't mean that there does not exist a zero off the line –  zerosofthezeta Sep 14 '13 at 9:00
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