What does the numerically verified part of the Riemann Hypothesis tell about prime numbers? - MathOverflow

What does the numerically verified part of the Riemann Hypothesis tell about prime numbers?

2012-10-28T21:57:34Z

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?

Answer by Stopple for What does the numerically verified part of the Riemann Hypothesis tell about prime numbers?

2012-10-29T03:42:45Z

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

http://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle

Answer by Stopple for What does the numerically verified part of the Riemann Hypothesis tell about prime numbers?

2012-10-29T15:35:21Z

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 $7.95\times 10^{12}$

Answer by Dimitris Koukoulopoulos for What does the numerically verified part of the Riemann Hypothesis tell about prime numbers?

2012-10-29T16:37:01Z

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 $x$.