Density Intervals of Zeros of the Partial Sums of the Dirichlet Eta Function

Abstract

It is shown that the set Rn := {Rz : ηn(z) = 0} contains an interval [αn, bn] for some αn < 0 and 0 < bn := sup{Rz : ηn(z) = 0}, where ηn(z) := Σn j=1(−1)j−1/jz is the nth, n > 2, partial sum of the Dirichlet eta function η(z) := Σ∞ j=1(−1)j−1/jz. It means that in the strip [αn, bn]×R no vertical sub-strip is zero-free for ηn(z), n > 2. Since lim infn→∞ bn ≥ 1, that property is, in particular, asymptotically true for the partial sums ηn(z) in the critical strip (0, 1) × R.