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A general strong Nyman-Beurling criterion for the Riemann hypothesis. (English) Zbl 1119.11048
The author considers the Müntz operator \[ g(x)=Pf(x):=\sum_{n\geq1}f(nx)-\frac{1}{x}\int_0^\infty f(t)\,dt \] for suitable \(f\). For certain \(f\) with both \(f,g\in L_2(0,\infty)\) it is true that the famous Riemann Hypothesis (all complex zeros of the Riemann zeta-function \(\zeta(s)\) have real parts equal to 1/2) is equivalent to the fact that \(f\) is in the \(L_2\) closure of the vector space generated by the dilatations \(g(kx)\) when \(k\in\mathbb N\). In this paper the author obtains, by methods from functional analysis, additional equivalent statements of the Riemann Hypothesis. An essential rôle is played by the classical representation \[ \zeta(s)=s\int_0^\infty\frac{[x]-x}{x^{s+1}}\,dx\qquad(0<\sigma<1,\;s=\sigma+it), \] where \([x]\) is the integer part of \(x\). This is a special case of the Müntz formula, which says that under suitable conditions \[ \zeta(s){\widehat f}(s) =\widehat {Pf}(s)\qquad(0<\sigma<1,\;s=\sigma+it), \] where \({\widehat f}(s)=\int_0^\infty x^{s-1}f(x)\,dx\) is the usual Mellin tranform of \(f\).

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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