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

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Matiyasevich Y., Saidak F., Zvengrowski P.

Acta Arithmetica

2014

tom 166

nr 2

189-200

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.

Ograniczanie wyników

1 Acta Arithmetica

1 Matiyasevich Y.

1 Saidak F.

1 Zvengrowski P.

1 2014

Wyniki wyszukiwania

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