Remarks on Weil's quadratic functional in the theory of prime numbers. I
Weil's explicit formula, in the case of the Riemann zeta-function \(\zeta(s)\), can be stated as follows. Let \(f\in C_0^{\infty}((0,\infty))\) be a smooth complex-valued function with compact support in \((0,\infty)\), let \(\widetilde{f}(s)=\int_0^{\infty} f(x)x^{s-1} dx\) be the Mellin transform of \(f\) and \(f^*(x)=\frac 1{x}f(\frac 1{x})\). Then \[ \sum_{\rho} \widetilde f(\rho) =\int_0^{\infty} f(x) dx + \int_0^{\infty} f^*(x) dx -\sum_{n=1}^{\infty} \Lambda(n)\{f(n)+f^*(n)\} - \] \[ -(\log 4\pi +\gamma)f(1)-\int_1^{\infty} \biggl\{f(x)+f^*(x)- \frac{2}{x}f(1)\biggr\} \frac{x dx}{x^2-1}, \] where the sum on the left hand side is over all complex zeros of \(\zeta(s)\) and \(\gamma\) is Euler's constant. Denote by \(\mathcal T(f)\) the right hand side of the above explicit formula. \(\mathcal T(f)\) is a linear functional and \(\mathcal T(f*\overline{f}^*)\) is called Weil's quadratic functional, where ``\(*\)'' stands for the multiplicative convolution of functions. \textit{A. Weil} [Meddel. Lunds Univ. Mat. Sem. (dedié à M. Riesz), 252-265 (1952; Zbl 0049.03205)] showed that \(\mathcal T(f*\overline{f}^*)\) is positive semidefinite if and only if the Riemann Hypothesis (R.H.) is true. This paper studies Weil's functional from a variational point of view, restricted to two classes of Hilbert spaces, sufficiently wide for testing the validity of the R.H. It is proved that the infimum of \(\mathcal T(f*\overline{f}^*)\) is attained in the unit sphere of the \(L^2\)-space of functions with support in a given interval \([-t,t]\). Furthermore, let \(W_0^{1,2}([M^{-1},M])\) be the space of functions \(f\) with \(f, Df\in L^2((0,\infty))\) and compact support in \([M^{-1},M]\), with norm \[ \|f\|'=\left( \int_{M^{-1}}^M |Df(x)|^2 \right) ^{1/2} dx, \] where \(D\) is the translation invariant differential operator \(D = x(d/dx)\). If the infimum of \(\mathcal T(f*\overline{f}^*)\) in the unit sphere of the space \(W_0^{1,2}([M^{-1},M])\) is negative, then this infimum is attained. The Fourier transform of this functional leads to a quadratic form in infinitely many variables. Finite truncations of it and the corresponding eigenvalues are investigated. It is shown that if the R.H. is false but only with finitely many non-trivial zeros off the critical line, then the number of negative eigenvalues is exactly \(1/2\) of the number of zeros failing to satisfy the R.H., assuming that the truncation is large enough.
scientific article; zbMATH DE number 1846845