Thursday, August 1, 2013

1307.8395 (Guilherme França et al.)

Statistical and other properties of Riemann zeros based on an explicit
formula for the n-th zero

Guilherme França, André LeClair
Very recently, a transcendental equation satisfied by the $n$-th Riemann zero on the critical line was derived by one of us. Here we provide a more detailed analysis of this result, demonstrating more rigorously that the Riemann zeros occur on the critical line with real part equal to 1/2, and their imaginary parts satisfy such a transcendental equation. From this equation, the counting of zeros on the critical line can be derived, yielding precisely the Riemann-von Mangoldt counting function $N(T)$ for the zeros on the entire critical strip. Therefore, these results constitute a proposal for establishing the validity of the Riemann Hypothesis. We also obtain an approximate solution of the transcendental equation, in closed form, based on the Lambert $W$ function. This yields a very good estimate for the Riemann zeros, for arbitrarily high values on the critical line. We then obtain numerical solutions of the complete transcendental equation, yielding accurate numbers for the Riemann zeros, which agree with previous known results found in the literature. Employing these numerical solutions, we verify that they are accurate enough to confirm Montgomery's pair correlation conjecture and also to reconstruct the prime number counting formula.
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