On the Ratio of the Modulus of Dirichlet Eta Function Values of Critical Line Symmetrical Arguments and the Riemann Hypothesis
چکیده
ABSTRACT. Said Sn(s) the nth partial sum of the Dirichlet η function, Rn(s) the corresponding remainder term, and ρ = 2 +α + it , τ = 1 2 −α + it , a pair of arguments symmetrical with respect to the critical line, and restricting the following considerations to the domain corresponding to the interior of the critical strip, it is shown that proving the sequence of functions {|Sn(ρ)|/|Sn(τ)|} to be uniformly convergent would confirm the validity of the Riemann Hypothesis. In fact, in such case the following inequality limn→∞ |Sn(ρ)|/|Sn(τ)| > limn→∞ |Rn(ρ)|/|Rn(τ)| would hold. However, making the additional hypothesis that said pair of critical line symmetrical arguments also corresponds to a pair of zeros would instead result in limn→∞ |Sn(ρ)|/|Sn(τ)| = limn→∞ |Rn(ρ)|/|Rn(τ)| = 0 . This contradicting result would hence suggest that there cannot be zeros of η lying off the critical line. Said hypothesis of uniform convergence represents in fact a novel equivalent reformulation of the RH.