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Linear relations of zeroes of the zeta-function


Authors: D. G. Best and T. S. Trudgian
Journal: Math. Comp. 84 (2015), 2047-2058
MSC (2010): Primary 11M26; Secondary 11M06
DOI: https://doi.org/10.1090/S0025-5718-2014-02916-5
Published electronically: December 29, 2014
MathSciNet review: 3335903
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Abstract: This article considers linear relations between the nontrivial zeroes of the Riemann zeta-function. The main application is an alternative disproof of Mertens' conjecture by showing that $ \limsup _{x\rightarrow \infty } M(x) x^{-1/2} \geq 1.6383$, and $ \liminf _{x\rightarrow \infty } M(x) x^{-1/2} \leq -1.6383.$


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Additional Information

D. G. Best
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, AB T1K 3M4, Canada
Address at time of publication: School of Mathematical Sciences, Monash University, Clayton, VIC 3168, Australia
Email: darcy.best@monash.edu

T. S. Trudgian
Affiliation: Mathematical Sciences Institute, The Australian National University, ACT 0200, Australia
Email: timothy.trudgian@anu.edu.au

DOI: https://doi.org/10.1090/S0025-5718-2014-02916-5
Keywords: Riemann zeta-function zeros, linear relations of ordinates
Received by editor(s): August 18, 2013
Received by editor(s) in revised form: November 14, 2013
Published electronically: December 29, 2014
Additional Notes: The first author was supported by NSERC CGS-M and Alberta Innovates – Technology Futures.
The second author was supported by ARC Grant DE120100173.
Article copyright: © Copyright 2014 American Mathematical Society