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A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis. (English) Zbl 1218.11079
Riemann \(\xi\) function \(\xi(s)\) is defined as the product \[ \xi(s)=\frac12s(s-1)\pi^{-\frac12s} \Gamma\left(\frac12s\right)\zeta(s), \] where \(\zeta(s)\) is the Riemann zeta-function and \(\Gamma(s)\) 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)]

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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[1] H. Davenport, Multiplicative Number Theory, 2nd ed., revised by H. L. Montgomery, Graduate Texts in Mathematics 74, Springer-Verlag, New York – Berlin, 1980. · Zbl 0453.10002
[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. · Zbl 1142.11001
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