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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 \(L(s, \chi)\)
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