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Riemann sums and Möbius. (English) Zbl 1442.11014

Summary: Let \(S\) be the set of square-free natural numbers. A Hilbert-Schmidt operator, \(\mathcal A\), associated to the Möbius function has the property that it maps from \( \bigcup_{0 < r < \infty}l^r(S)\) to \( \bigcap_{0 < r < \infty}l^r(S)\), injectively. If \(0<r<2\) and \(\xi\in l^r(S)\), the series \({f_\zeta} = \sum\nolimits_{n \in s} \mathcal A\zeta (x)\cos2\pi nx \) converges uniformly to an element of \(f_\xi\mathcal R_0\), i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} f_\zeta (j/N) = 0\), \(N = 1,2,\dots\).
If \(\mathcal A\zeta_{\lambda} = \lambda \zeta_\lambda\), \(\zeta_\lambda(1) = 1\), then \(\xi_\lambda\) is multiplicative. If \(f_{\zeta_\lambda} \in \Lambda_a\), the space of \(\alpha\)-Lipschitz continuous functions, for some \(\alpha >0\), and if \(\chi\) is any Dirichlet character, then \(L(s, \chi) \ne 0\), \(\operatorname{Re} s>1-\alpha\).
Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to \(f_\xi\in\Lambda_\alpha\), \(\alpha <1/2\), \(\xi\in l^r(S)\), \(0<r<2\). Using a 1991 estimate by R. C. Baker and G. Harman [J. Lond. Math. Soc., II. Ser. 43, No. 2, 193–198 (1991; Zbl 0685.10027)], one finds GRH implies
\[ f_\xi\in\Lambda_\alpha, \quad \alpha <1/4,\ \xi\in l^r(S), \ 0<r<2.\]
The question of whether \(\mathcal R_0\cap\Lambda_\alpha\neq\{0\}\) for some positive \(\alpha >0\) is open.
[William A. Veech passed away on August 30, 2016, before the final revisions were made to his paper. The Editorial Board of Journal d’Analyse Mathématique thanks Giovanni Forni and Jon Fickenscher for helping to prepare the paper for publication.]

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Citations:

Zbl 0685.10027
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References:

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