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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:
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