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A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis. (English) Zbl 1218.11079

Riemann ξ function ξ(s) is defined as the product

ξ(s)=1 2s(s-1)π -1 2s Γ1 2sζ(s),

where ζ(s) is the Riemann zeta-function and Γ(s) is the Euler gamma function. The authors prove that the xi function is strictly increasing in modulus along every horizontal half-line lying in any open right half-plane that contains no zeros of xi. Similarly, the modulus strictly decreases on each horizontal half-line in any zero-free, open left half-plane. Under the Riemann hypothesis this statement appears as Exercise 1 (e) in Section 13.2 of H. L. Montgomery and R. C. Vaughan [Multiplicative Number Theory. I. Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press (2007; Zbl 1142.11001)]

11M26Nonreal zeros of ζ(s) and L(s,χ); Riemann and other hypotheses
[1]H. Davenport, Multiplicative Number Theory, 2nd ed., revised by H. L. Montgomery, Graduate Texts in Mathematics 74, Springer-Verlag, New York – Berlin, 1980.
[2]H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I, Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.