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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 $$f$$.

##### MSC:
 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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