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Epreuve numérique d’une supposition de P. Turán. (French) Zbl 1145.11324
Let \(L_s(n)=\sum_{m=1}^n\lambda(m)m^{-s}\), where \(\lambda(m)\) is Liouville’s function (\(\lambda(m)=(-1)^\mu\) when \(m\) is a product of \(\mu\) equal or distinct primes). P. Turán conjectured that \(L_1(n)>0\) \((n\geq 1)\) [Danske Vid. Selsk. Mat.-Fys. Medd. 24, No. 17, 36 p. (1948; Zbl 0031.30204)]. Numerical evidence in support of this is here given for the range \(1000<n\leq 4528\). The author also reports that he has repeated previous calculations up to \(n=1000\) and found minor discrepancies (in the fifth place of decimals). The least value found is \(L_1(2837)=0.0002393\). Turán’s conjecture is on much the same footing as Pólya’s conjecture \(L_0(n)\leq 0\) \((n\geq 2)\), and is open to similar doubts.

11M06 \(\zeta (s)\) and \(L(s, \chi)\)