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Journal of the London Mathematical Society 2006 73(2):399-414; doi:10.1112/S0024610706022599
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© The London Mathematical Society

A Li-Type Criterion for Zero-Free Half-Planes of Riemann's Zeta Function

Pedro Freitas

Faculdade de Motricidade Humana (TU Lisbon) and Mathematical Physics Group of the University of Lisbon Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal freitas{at}cii.fc.ul.pt

Received 25 November 2004. Revision received 2 June 2005.

We define a sequence of real functions which coincide with Li's coefficients at one and which allow us to extend Li's criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip 0 < R(z) < 1. We study some of the properties of these functions, including their oscillatory behaviour.


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