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

Yu. Matiyasevich, F. Saidak, and P. Zvengrowski. "Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions." Acta Arithmetica 166.2 (2014): 189-200. <http://eudml.org/doc/279755>.

@article{Yu2014,
abstract = { 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. },
author = {Yu. Matiyasevich, F. Saidak, P. Zvengrowski},
journal = {Acta Arithmetica},
keywords = {Riemann zeta function; Dirichlet -functions; monotonicity},
language = {eng},
number = {2},
pages = {189-200},
title = {Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions},
url = {http://eudml.org/doc/279755},
volume = {166},
year = {2014},
}

TY - JOUR
AU - Yu. Matiyasevich
AU - F. Saidak
AU - P. Zvengrowski
TI - Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions
JO - Acta Arithmetica
PY - 2014
VL - 166
IS - 2
SP - 189
EP - 200
AB - 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.
LA - eng
KW - Riemann zeta function; Dirichlet -functions; monotonicity
UR - http://eudml.org/doc/279755
ER -