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ON THE RATIO OF THE MODULUS OF DIRICHLET ETA FUNCTION VALUES OF CRITICAL LINE SYMMETRICAL ARGUMENTS AND THE RIEMANN HYPOTHESIS.

By Luca Ghislanzoni

Abstract

ABSTRACT. Said Sn(s) the nth partial sum of the Dirichlet η function, and Rn(s) the corresponding remainder term, by combining an analytic approach with a geometric one it is shown that inside the critical strip, for any pair of critical line symmetrical arguments, ρ = 1 2 + α + it and τ = 1 2 − α + it, the inequality limn→ ∞ |Sn(ρ)|/|Sn(τ) |> limn→ ∞ |Rn(ρ)|/|Rn(τ) | always holds. However, making the additional hypothesis that said pair of critical line symmetrical arguments corresponds to a pair of zeros would instead result in limn→ ∞ |Sn(ρ)|/|Sn(τ) | = limn→ ∞ |Rn(ρ)|/|Rn(τ) | = 0. This contradicting result suggests that in the interior of the critical strip there cannot be zeros of η lying off the critical line. 1

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