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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.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$