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Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function

Rusev, Peter (2010)

Fractional Calculus and Applied Analysis

AMS Subject Classification 2010: 11M26, 33C45, 42A38.Necessary and sufficient conditions for absence of zeros of ζ(s) in the half-plane σ ... Expansion of holomorphic functions in series of Hermite polynomials ...

Higher regularizations of zeros of cuspidal automorphic $L$ -functions of ${GL}_{d}$

Masato Wakayama, Yoshinori Yamasaki (2011)

Journal de Théorie des Nombres de Bordeaux

We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic $L$ -functions of ${GL}_{d}$ over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.

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

Yu. Matiyasevich, F. Saidak, P. Zvengrowski (2014)

Acta Arithmetica

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 <...

Hybrid joint universality theorem for Dirichlet L-functions

Łukasz Pańkowski (2010)