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Notes on the Riemann $$\zeta$$-function. IV. (Notes sur la fonction $$\zeta$$ de Riemann. IV.) (French) Zbl 1096.11032
This is the fourth paper in a series that deals with equivalents of the Riemann Hypothesis (RH) (that all complex zeros of $$\zeta(s)$$ satisfy $$\operatorname{Re} s = 1/2$$) [Part I: Adv. Math. 139, 310–321 (1998; Zbl 0920.11062), Part II: ibid. 143, No. 2, 284–287 (1999; Zbl 0937.11032) and Part III: ibid. 149, 130–144 (2000; Zbl 1008.11032)].
The central theme is L. Báez-Duarte’s equivalent criterion [Atti Acad. Naz. Lincei Rend. Mat. Appl. 14, 5–11 (2003; Zbl 1097.11041)] of the RH that, in the authors’ reformulation, $\lim_{n\to\infty}d^2({\mathbf 1}, E_n) = 0.(1)$ Here $$H:= \ell^2(\mathbb N^*,1/(k^2+k))$$ is the Hilbert space of complex sequences $${\mathbf x} = (x_k)_{k\geq1}$$ with the scalar product defined as
$\langle {\mathbf x}, {\mathbf y}\rangle \,:= \sum_{k=1}^\infty {x_k{\bar y}_k\over k(k+1)},$
and $$\langle{\mathbf x}, {\mathbf x}\rangle \;<\; +\infty$$. By $$\{u\}$$ we denote the fractional part of $$u$$, $${\pmb \pi_a}$$ is the sequence defined by $${\pmb \pi_a}(k) = a\{k/a\}$$, and $$E_n$$ is the vector subspace of $$H$$ generated by $${\pmb \pi_2}, {\pmb \pi_3},\ldots, {\pmb \pi_n}$$. Finally $${\mathbf 1}$$ denotes the sequence equal to 1 and $$d$$ the distance induced by the norm of $$H$$. The authors raise several questions on this subject, explain why it is plausible to conjecture that the expression in (1) is asymptotic ($$\gamma = -\Gamma'(1)$$) to
${2+\gamma - \log(4\pi)\over \log n}\,\qquad(n\to\infty),$ and prove two interesting theorems. The first result is that $d^2((I-S)({\mathbf 1}),(I-S)(E_n)) = A,$ where they evaluate explicitly the constant $$A>0$$ in terms of the Euler function and a closely related arithmetic function, where the result holds in the space $$\ell^2((\mathbb N^*,1/(\zeta(2)k^2))$$, $$I$$ being the identity operator and $$S$$ the operator which sends $$(x_1,x_2,x_3,\ldots)$$ to $$(0,x_1,x_2,\ldots)$$.
The second result is that the limit (1) holds in another, precisely defined functional space, and the value of the limit is again explicitly evaluated. Some of the questions raised by the authors have been settled in the meantime, which is indicated at the end of the paper.

##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Citations:
Zbl 0920.11062; Zbl 0937.11032; Zbl 1008.11032; Zbl 1097.11041
Full Text:
##### References:
 [1] Aitken, A.C., Determinants and matrices, (1956), Oliver and Boyd Edinburgh · Zbl 0022.10005 [2] Báez-Duarte, L., New versions of the nyman-Beurling criterion for the Riemann hypothesis, Internat. J. math. math. sci., 31, 387-406, (2002) · Zbl 1069.11037 [3] L. Báez-Duarte, A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, 2, à paraı̂tre dans Atti Acad. Naz. Lincei Rend. Mat. Appl, et : . · Zbl 1097.11041 [4] Báez-Duarte, L.; Balazard, M.; Landreau, B.; Saias, E., Notes sur la fonction ζ de Riemann, 3, Adv. in math., 149, 130-144, (2000) · Zbl 1008.11032 [5] L. Báez-Duarte, M. Balazard, B. Landreau, E. Saias, Étude de l’autocorrélation multiplicative de la fonction $$«$$partie fractionnaire$$»$$, à paraı̂tre au Ramanujan Journal. · Zbl 1173.11343 [6] Balazard, M.; Saias, E., Notes sur la fonction ζ de Riemann, 1, Adv. in math., 139, 310-321, (1998) · Zbl 0920.11062 [7] Balazard, M.; Saias, E., The nyman-Beurling equivalent form for the Riemann hypothesis, Expo. math., 18, 131-138, (2000) · Zbl 0954.11029 [8] Beurling, A., A closure problem related to the Riemann zeta-function, Proc. nat. acad. sci., 41, 312-314, (1955) · Zbl 0065.30303 [9] Burnol, J.-F., A lower bound in an approximation problem involving the zeros of the Riemann zeta function, Adv. in math., 170, 56-70, (2002) · Zbl 1029.11045 [10] Haukkanen, P.; Wang, J.; Sillanpää, J., On Smith’s determinant, Linear algebra appl., 258, 251-269, (1997) · Zbl 0883.15002 [11] Titchmarsh, E.C., The theory of the Riemann zeta-function (revised by D.R. heath-Brown), (1986), Clarendon Press Oxford · Zbl 0601.10026 [12] Vassiounine, V.I., Sur un système biorthogonal relié à l’hypothèse de Riemann (en russe), Alg. i an., 7, 118-135, (1995), (traduction anglaise dans St-Petersburg Math. J. 7 (1996) 405-419)
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