Remarks on Weil’s quadratic functional in the theory of prime numbers, I

Bombieri, Enrico

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 3, pp. 183-233

Résumé

This Memoir studies Weil’s well-known Explicit Formula in the theory of prime numbers and its associated quadratic functional, which is positive semidefinite if and only if the Riemann Hypothesis is true. We prove that this quadratic functional attains its minimum in the unit ball of the $L^{2}$-space of functions with support in a given interval $\left[ -t,t \right]$, and prove again Yoshida’s theorem that it is positive definite if $t$ is sufficiently small. The Fourier transform of the functional gives rise to a quadratic form in infinitely many variables and we then study its finite truncations and corresponding eigenvalues. In particular, if the Riemann Hypothesis is false but only with finitely many non-trivial zeros off the critical line we show that the number of negative eigenvalues is precisely one-half of the number of zeros failing to satisfy the Riemann Hypothesis, provided the truncation is big enough.