Abstract
We prove that Riemann’s xi function is strictly increasing (respectively, strictly decreasing) in modulus along every horizontal half-line in any zero-free, open right (respectively, left) half-plane. A corollary is a reformulation of the Riemann Hypothesis.
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H. Davenport, Multiplicative Number Theory, 2nd ed., revised by H. L. Montgomery, Graduate Texts in Mathematics 74, Springer-Verlag, New York — Berlin, 1980.
H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I, Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.
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Communicated by Attila Pethő
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Sondow, J., Dumitrescu, C. A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis. Period Math Hung 60, 37–40 (2010). https://doi.org/10.1007/s10998-010-1037-3
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DOI: https://doi.org/10.1007/s10998-010-1037-3