Mora, Gaspar
Density Intervals of Zeros of the Partial Sums of the Dirichlet Eta Function
Mediterranean Journal of Mathematics. 2018, 15:208. doi:10.1007/s00009-018-1252-3
URI: http://hdl.handle.net/10045/82174
DOI: 10.1007/s00009-018-1252-3
ISSN: 1660-5446 (Print)
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.
Keywords:Zeros of the partial sums of the eta function, Exponential polynomials, Diophantine approximation
Springer International Publishing
info:eu-repo/semantics/article