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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn064, 23 pages, doi:10.1093/imrn/rnn064 published on June 17, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

The Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line

Haseo Ki

Department of Mathematics, Yonsei University, Seoul 120-749, Korea

Correspondence: Correspondence to be sent to: haseo{at}yonsei.ac.kr

We study the horizontal distribution of zeros of {zeta}'(s) which are denoted as {rho}' =β' +i{gamma}'. We assume the Riemann hypothesis which implies β' ≥ 1/2 for any nonreal zero {rho}', equality being possible only at a multiple zero of {zeta} (s). In this paper, we prove that lim inf (β' –1/2)log {gamma}' != 0 if, and only if, for any c > 0 and s = {sigma} + it with 0 ≤|{sigma} -1/2| ≤c/log t (t>t0(c)), we have


Formula 1

where Formula is the zero of {zeta} closest to s (and to the origin, if there are two such). We also show that if lim inf (β'-1/2)log {gamma}' != 0, then for any c>0 and s={sigma} + it (t>t1(c)), we have


Formula 2

uniformly for 1/2+c/log t≤ {sigma} ≤ {sigma} 1 <1.



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This Article
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