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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.

MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11A25 Arithmetic functions; related numbers; inversion formulas
Citations:
Zbl 0065.30303; Zbl 0569.46011
Full Text:
References:
  Báez-Duarte, L., On Beurling’s real variable reformulation of the Riemann hypothesis, Adv. maths., 101, 10-30, (1993) · Zbl 0795.11035  Bercovici, H.; Foias, C., A real variable restatement of Riemann’s hypothesis, Israel J. maths., 48, 57-68, (1984) · Zbl 0569.46011  Beurling, A., A closure problem related to the Riemann zeta-function, Proc. nat. acad. sci., 41, 312-314, (1955) · Zbl 0065.30303  A. Blanchard, Initiation à la théorie analytique des nombres premiers, Dunod, 1969 · Zbl 0198.37602  Donoghue, W.F., Distributions and Fourier transforms, Pure appl. math., 32, (1969), Academic Press San Diego · Zbl 0188.18102  Helson, H., Convergence of Dirichlet series, L’analyse harmonique dans le domaine complexe, (1973), Springer Lecture Notes, p. 153-160  Levinson, N., Gap and density theorems, A.M.S. coll. publ. XXVI, (1940) · JFM 66.0332.01  Nikolski, N., Distance formulae and invariant subspaces, with an application to localization of zeroes of the Riemannζ, Ann. inst. Fourier, 45, 143-159, (1995) · Zbl 0816.30026  Nyman, B., On some groups and semigroups of translations, Thèse, Uppsala, (1950)  Paley, R.E.A.C.; Wiener, N., The Fourier transform in the complex domain, A.M.S. coll. publ. XIX, (1934) · Zbl 0006.25704  Schwartz, L., Etude des sommes d’exponentielles, (1959), Hermann Paris  Szász, O., Über die approximation stetiger funktionen durch lineare aggregate von potenzen, Math. ann., 77, 482-496, (1916) · JFM 46.0419.03  Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres, Cours spécialisé S.M.F., 1, (1996) · Zbl 0788.11001  Titchmarsh, E.C., The theory of the Riemann zeta-function (revised by D. R. heath-Brown), (1986), Clarendon Press Oxford · Zbl 0601.10026  Vassiounine, V.I., Sur un système biorthogonal relié à l’hypothèse de Riemann (en russe), Algebra i annaliz, 7, 118-135, (1995)
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