Riemann hypothesis
Turán's result

Pál Turán (1948) showed that if the functions ∑ n = 1 N n − s {\displaystyle \sum _{n=1}^{N}n^{-s}} have no zeros when the real part of s is greater than one then T (x) = ∑ n ≤ x λ (n) n ≥ 0 for x > 0, {\displaystyle T(x)=\sum _{n\leq x}{\frac {\lambda (n)}{n}}\geq 0{\text{ for }}x>0,} where λ(n) is the Liouville function given by (−1) if n has r prime factors. He showed that this in turn would imply that the Riemann hypothesis is true. But Haselgrove (1958) proved that T (x) is negative for infinitely many x (and also disproved the closely related Pólya conjecture), and Borwein, Ferguson & Mossinghoff (2008) showed that the smallest such x is 72 185 376 951 205. Spira (1968) showed by numerical calculation that the finite Dirichlet series above for N =19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+ N for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but Montgomery (1983) showed that for all sufficiently large N these series have zeros with real part greater than 1 + (log log N)/(4 log N). Therefore, Turán's result is vacuously true and cannot help prove the Riemann hypothesis.
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