Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions

• Autorzy: Yu. Matiyasevich , F. Saidak , P. Zvengrowski
• Języki publikacji: EN

Abstrakty

EN

As usual, let s = σ + it. For any fixed value of t with |t| ≥ 8 and for σ < 0, we show that |ζ(s)| is strictly decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality related to the monotonicity of all three functions is proved:
ℜ (η'(s)/η(s)) < ℜ (ζ'(s)/ζ(s)) < ℜ (ξ'(s)/ξ(s)).
It is also shown that extending the above monotonicity result for |ζ(s)|, |ξ(s)|, or |η(s)| from σ < 0 to σ < 1/2 is equivalent to the Riemann hypothesis. Similar monotonicity results will be established for all Dirichlet L-functions L(s,χ), where χ is any primitive Dirichlet character, as well as the corresponding ξ(s,χ) functions, together with the relation of this to the generalized Riemann hypothesis. Finally, these results will be interpreted in terms of the degree 1 elements of the Selberg class.

Kategorie tematyczne

• 11M06: ζ ( s ) and L ( s , χ )
• 11M26: Nonreal zeros of ζ ( s ) and L ( s , χ ) ; Riemann and other hypotheses

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 166
• Numer: 2
• Strony: 189-200

Daty

wydano

2014

Twórcy

autor

Yu. Matiyasevich

• Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg Department, (POMI RAN), 27, Fontanka, St. Petersburg, 191023, Russia

autor

F. Saidak

• Department of Mathematics, University of North Carolina, Greensboro, NC 27402, U.S.A.

autor

P. Zvengrowski

• Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Identyfikatory

DOI

10.4064/aa166-2-4