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Notes on the Riemann zeta-function. I. (Notes sur la fonction \(\zeta\) de Riemann. I.) (French) Zbl 0920.11062
Let \(S_n(t)=\sum^n_{k=1}\mu(k)\{{1\over kt}\}\) and \(B_n(t)= S_n (t)-n\{{1\over nt}\} \sum^n_{k=1} {\mu(k)\over k}\). The authors prove the following extension of theorems of A. Beurling [Proc. Natl. Acad. Sci. USA 41, 312-314 (1955; Zbl 0065.30303)] and H. Bercovici and C. Foias, Isr. J. Math. 48, 57-68 (1984; Zbl 0569.46011)]. For every real \(p\geq 1\) the following three statements are equivalent
a) The Riemann zeta function does not vanish in the half plane \(\text{Re} s>1/p\).
b) The sequence of functions \((B_n)\) converges to \(-1\) in \(L^r(0,1)\) for every \(r\) with \(1<r<p\).
c) The sequence of functions \((S_n)\) is bounded in \(L^r(0,1)\) for every \(r\) with \(1<r<p\).
The proof follows standard arguments used for \(\zeta(s)\) and sums involving the Möbius function. Several other related questions are discussed.

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11A25 Arithmetic functions; related numbers; inversion formulas
Full Text: DOI
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