The Liouville Function and the Riemann Hypothesis

Michael J. Mossinghoff () and Timothy S. Trudgian ()
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Michael J. Mossinghoff: Davidson College, Department of Mathematics and Computer Science
Timothy S. Trudgian: The University of New South Wales Canberra, School of Physical, Environmental and Mathematical Sciences

A chapter in Exploring the Riemann Zeta Function, 2017, pp 201-221 from Springer

Abstract: Abstract For nearly a century, mathematicians have explored connections between the Liouville function and the Riemann hypothesis. We describe a number of connections regarding oscillations in sums involving the Liouville function, including L 0(x) = ∑ n ≤ x λ(n), first studied by Pólya, and L 1(x) = ∑ n ≤ x λ(n)∕n, explored by Turán. We establish new lower bounds on the size of the oscillations in such sums. In particular, we prove that L 0 ( x ) > x $$L_{0}(x)> \sqrt{x}$$ infinitely often, show that L 1 ( x )

Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-59969-4_9

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DOI: 10.1007/978-3-319-59969-4_9

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