Variations on criteria of Pólya and Turán for the Riemann hypothesis

Abstract

For 0≤α≤1, letLα(x)=∑n≤xλ(n)nα, where λ(n) is the Liouville function. Then famous criteria of Pólya and Turán claim that the eventual sign constancy of each of L0(x) and L1(x) alone implies the Riemann hypothesis. However, Haselgrove disproved the eventual sign constancy hypothesis. As a remedy for this, we show that the eventual negativity of each of Lα(x), Lα′(x), Lα″(x) for all 1/2<α<1 is equivalent to the Riemann hypothesis. Analogous equivalent conditions for the Riemann hypothesis are formulated in terms of partial sums of the Möbius function as well. Our criteria indicate a new tendency concerning the values of the Liouville and Möbius functions in stark contrast with their random behavior. Further generalizations are given in two essential directions, namely that equivalent criteria are developed for any quasi-Riemann hypothesis, and we allow the partial sums to be supported on semigroups after sifting out integers divisible by members of a set of prime numbers whose size is under control. Our negativity conditions for the quasi-Riemann hypothesis exhibit a curious rigidity in two respects, first it doesn't matter which particular primes we use to sift out, but only the size of the set, second the sign of the partial sums is unaltered with respect to two successive derivatives. Finally, other competing necessary or sufficient conditions for a quasi-Riemann hypothesis are studied regarding the monotonicity, zeros and sign changes of partial sums of Dirichlet series.