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

Bombieri, Enrico. "Remarks on Weil’s quadratic functional in the theory of prime numbers, I." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.3 (2000): 183-233. <http://eudml.org/doc/252338>.

@article{Bombieri2000,
abstract = {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.},
author = {Bombieri, Enrico},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Prime number theory; Riemann Hypothesis; Explicit Formula; Weil's quadratic functional; Riemann zeta function; Fourier transform; quadratic form; eigenfunction},
language = {eng},
month = {9},
number = {3},
pages = {183-233},
publisher = {Accademia Nazionale dei Lincei},
title = {Remarks on Weil’s quadratic functional in the theory of prime numbers, I},
url = {http://eudml.org/doc/252338},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Bombieri, Enrico
TI - Remarks on Weil’s quadratic functional in the theory of prime numbers, I
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/9//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - 3
SP - 183
EP - 233
AB - 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.
LA - eng
KW - Prime number theory; Riemann Hypothesis; Explicit Formula; Weil's quadratic functional; Riemann zeta function; Fourier transform; quadratic form; eigenfunction
UR - http://eudml.org/doc/252338
ER -