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I'm curious about the following question:

As of 2005(?) the Riemann hypothesis is verified for the first 10 trillion zeroes, they are all on the critical line. Does this verification gives us any information about prime number?

In particular, are there any results saying if all the non-trivial zeroes whose imaginary part is < N and > 0 are on the critical line, then we understand something about prime number < M, where M is a number depend on N?

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The magic words are "explicit formula". The short answer is "no". –  Igor Rivin Oct 28 '12 at 23:29
Per terrytao.wordpress.com/2012/02/01/…, it tells us that every odd integer larger than 1 is the sum of at most five primes. –  Will Sawin Oct 28 '12 at 23:42
Knowing the location of first 2000 or so zeros of the zeta-function above the real axis to 75 digits of accuracy seems to have been essential in Odlyzko and te Riele's disproof of the Merten's conjecture. See: oai.cwi.nl/oai/asset/1823/1823A.pdf –  Micah Milinovich Oct 29 '12 at 0:15
@Will, well, it tells us every odd integer larger than one up to a certain 'ceiling' is such a sum. The rest of the odd integers are dealt with by other methods that only work above that ceiling. –  David Roberts Oct 29 '12 at 4:06
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3 Answers

If you look at the explicit formula, then you can get a bound for the error term in the PNT: If $$ \psi(x) = \sum_{p^k\le x}\log p, $$ then formula (9) in page 109 of Davenport's book (multiplicative number theory) implies that $$ \psi(x) = x + \sum_{-T\le \gamma\le T}\frac{x^\rho}{\rho} + O\left(1+\frac{ x\log^2(xT) }{T}\right), $$ for every $T\ge1$, where the implied constant is completely effective. Now, if we know that all the zeroes of $\zeta$ up to height $T$ lie on the critical line, then this automatically implies that $$ |\psi(x) - x | \le x^{1/2}\sum_{-T\le \gamma\le T} \frac{1}{\sqrt{1/4+\gamma^2}} + O\left(1+ \frac{ x\log^2(xT)}{T} \right) \ll x^{1/2}\log^2T + \frac{ x\log^2(xT)}{T}, $$ for some effective implied constants. So, in certain ranges of $x$, depending on $T$, you can get very good bounds on the size of $\psi(x)$ and therefore on how many primes there are up to $x$.

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To put it a bit more loosely (ignoring the log factors): if one has verified RH up to height $T_0$, then one can accurately count primes in intervals of the form $[x,x+y]$, so long as $y \gg \max( x/T_0, x^{1/2} )$, by optimising the above bound in $T \in [1,T_0]$. So numerical RH up to height $T_0$ is roughly "as good as" full RH for counting primes up to about $T_0^2$, and gives a partial substitute for RH beyond that scale which becomes increasingly strong as $T_0$ increases. –  Terry Tao Oct 29 '12 at 17:24
p.s. the paper of Ramare and Saouter ams.org/mathscinet-getitem?mr=1950435 managed to obtain a completely effective version of the above inequality that saves a logarithm and is useful for a number of effective analytic number theory purposes (as mentioned in comments, for instance, I used it to show every odd number up to $8.7 \times 10^{36}$ was the sum of at most five primes, and used some other arguments to cover other ranges). –  Terry Tao Oct 29 '12 at 17:37
$\gamma$ is the imaginary part of the zeroes right? –  36min Oct 30 '12 at 3:10
@36min: Yes, usually we write $\rho=\beta+i\gamma$ for a zero of the Riemann $\zeta$ function. –  Dimitris Koukoulopoulos Nov 1 '12 at 16:13
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The disproof of Mertens' conjecture (cited above) was certainly a computations tour de force using explicit values of the zeros of $\zeta(s)$. Another good example is the paper of Rosser and Schoenfeld "Sharper Bounds for the Chebyshev Functions $\theta(x)$ and $\psi(x)$" Math. Comp., v. 29 1975, pp. 243-269.

We know by the Prime Number Theorem that $\Psi(x)\sim x$. Rosser and Schoenfeld use values of zeros of $\zeta(s)$ to show, for example, that for $\log(x)>105$, we have $|\Psi(x)-x|<x\epsilon(x)$, where, for $X=(\log(x)/9.6459 08801)^{1/2}$ $$ \epsilon(x)= 0.257634 \left(1 + \frac{0.96642}{X} \right) X^{3/4}\exp(-X). $$ The paper contains a number of results of this flavor, about the Chebyshev function $\theta(x)$, and about asymptotics of the $n$th prime $p_n$.

The reason it is difficult to convert results about low lying zeros to results about small primes is that the Explicit Formula, (mentioned in comments above) has the primes and zeros lying on opposite sides of a Fourier Transform. The Heisenberg Uncertainty Principle applies


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Wait, the formula for $\epsilon(x)$ can't be right as written. The $\exp(-x)$ factor damps everything else, and would yield $|\Psi(x) - x| \ll x^{7/4} \exp(-x)$ for all $x$, an absurdly tight inequality. [Also, it's the Uncertainty Principle, not "Principal".] –  Noam D. Elkies Oct 29 '12 at 5:54
@Noam: Thanks, corrected via adding the definition in R&S of X as a function of x. –  Stopple Oct 29 '12 at 15:18
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Another application of the computation of large numbers of Riemann zeros (beyond verification of the Riemann Hypothesis) is towards bounding the deBruijn-Newman constant $\Lambda$:

deBruijn introduced a deformation parameter $t$ in the Riemann $\Xi$ function so that $\Xi_0(x)=\Xi(x)$ and the Riemann zeros $x(t)$ flow according to the "backward heat equation." Together their work shows the existence of a constant $\Lambda$ such that, for $\Lambda\le t$ the function $\Xi_t(x)$ has only real zeros, while for $t<\Lambda$ there exist complex zeros. The Riemann Hypothesis is the conjecture that $\Lambda\le 0$. Newman made the complementary conjecture that $\Lambda\ge 0$, writing "This new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Csordas, Smith, and Varga were able to analyze the ODEs governing the motion of the zeros, and use the fact that an very close pair of zeros, so-called Lehmer pairs, would give a lower bound on $\Lambda.$

The current best bound via this approach, due to Saouter, Gourdon, and Demichel, is that $$ -1.14\times 10^{-11}<\Lambda $$ based on a Lehmer pair at height about $7.95\times 10^{12}$

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