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

Abstract

As usual let s = σ + it. For any fixed value t = t 0 with |t 0 | ≥ 8, and for σ ≤ 0, we show that |ζ(s)| is strictly monotone decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality relating the monotonicity of all three functions is proved: 1 2000 Mathematics Subject Classification : Primary 11M06, Secondary 11M26

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

“…Proof. As remarked in [MSZ14], for a homolorphic function f , a simple calculation, using the Cauchy-Riemann equation, leads to…”

Monotonicity properties of ‐functions

“…A number of interesting results on the horizontal monotonicity of various Dirichlet series, for instance, for the Riemann zeta function and related L‐functions, have been known, some of which can be attributed to Spira , Saidak and Zvengrowski , Matiyasevich, Saidak and Zvengrowski , Zhang , and the references therein. Concerning horizontal monotonicity along the real line, Alzer proved the monotonicity properties of a function related to the Riemann zeta function given by .…”

On the Modulus of the Selberg Zeta-Functions in the Critical Strip

“…In the paper [11], it was proved the following relation between functions ζ(s) and ξ(s). Sondow and Dumitrescu proved in [17] the following theorem for the function ξ(s).…”

“…Later, Theorem 3 was reproved in [11] in a slightly different way. Related properties of the functions ζ(s) and ξ(s) in the critical strip were also investigated in [15].…”

“…Then |f (s)| is strictly decreasing with respect to σ in D, i.e., for each s 0 ∈ D, there exists δ > 0 such that |f (s)| is strictly monotonically decreasing with respect to σ on the horizontal interval from s 0 − δ to s 0 + δ. (b) Conversely, if |f (s)| is decreasing with respect to σ in D, then Re f (s) f (s) ≤ 0 for all s ∈ D such that f (s) = 0.The proof of the lemma is given in[11].Remark 1. Of course, the analogous results hold for monotonically increasing |f (s)| and Re f (s) f (s) > 0.…”