Monotonicity of Riemann Zeta Ratios and the Critical Strip

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Introduction

The Riemann zeta function, denoted by ζ(s), is the central object of study in analytic number theory. Since its formal introduction by Bernhard Riemann in 1859, its properties have been shown to be inextricably linked to the distribution of prime numbers. While the Riemann Hypothesis (RH) remains unsolved, significant progress has come from investigating the function’s behavior along the real axis and within the critical strip. The research paper arXiv:hal-04346907, authored by Mourad E. H. Ismail and Erik Koelink, provides a rigorous analysis of the monotonicity properties of the zeta function, specifically focusing on the ratio ζ(x+1)/ζ(x) for real values of x.

The motivation behind this study lies in the intersection of classical analysis and number theory. Ismail and Koelink investigate whether certain functions derived from ζ(s) possess the property of complete monotonicity. A function is said to be completely monotonic if its derivatives alternate in sign indefinitely. According to Bernstein’s Theorem, such a function is the Laplace transform of a non-negative measure. By establishing these properties, researchers can uncover hidden structural constraints on the function's growth and its zero distribution.

Mathematical Background

To understand the contribution of arXiv:hal-04346907, we define the core mathematical objects. The Riemann zeta function is defined for Re(s) > 1 by the Dirichlet series ζ(s) = Σ n^-s. The paper focuses on the function f(x) = ζ(x+1)/ζ(x) as an analytic object on the interval (1, ∞).

Definition: Complete Monotonicity

A function f(x) is completely monotonic on an interval if f is infinitely differentiable and satisfies (-1)ⁿ f⁽ⁿ⁾(x) ≥ 0 for all x and all non-negative integers n. This property is significant because the Riemann Hypothesis is equivalent to certain statements about the positivity of coefficients in the Taylor expansion of functions related to ζ(s). Complete monotonicity is the continuous analogue of such positivity conditions.

Main Technical Analysis

Spectral Properties and the Log-Zeta Derivative

The analysis in arXiv:hal-04346907 centers on the behavior of the function ψ(x) = -ζ'(x)/ζ(x). For x > 1, this can be represented as a Dirichlet series Σ Λ(n) n^-x, where Λ(n) is the von Mangoldt function. Taking the k-th derivative yields an expression where the sign is determined by (-1)^k+1. This implies that -ζ'(x)/ζ(x) is a completely monotonic function. Ismail and Koelink extend this logic to the ratio f(x) = ζ(x+1)/ζ(x) by linking its monotonicity to the differences of the ψ function.

The Hadamard Product and Zero Distribution

A critical aspect of the technical analysis involves the Weierstrass-Hadamard product for the zeta function. The monotonicity of the ratio ζ(x+1)/ζ(x) on the real line is influenced by the location of the non-trivial zeros ρ. If any zeros were to deviate significantly from the critical line Re(s) = 1/2, they would exert a pull on the derivatives of ζ(s) that could violate the complete monotonicity of the ratio on the real axis. Thus, the smoothness of the function for real x provides a baseline for the behavior of zeros in the complex plane.

Monotonicity and the von Mangoldt Function

The core of the argument rests on the observation that the coefficients Λ(n) are non-negative. The authors demonstrate that the derivative of the ratio ζ(x+1)/ζ(x) can be expanded into a series where the coefficients are combinations of Λ(n). The technical challenge addressed in arXiv:hal-04346907 is proving that these combinations remain signed correctly for all orders of differentiation, requiring a delicate handling of the convolution of the von Mangoldt function.

Novel Research Pathways

The Monotonicity Criterion for the Critical Strip: Extending the Ismail-Koelink property into the complex plane for s = σ + it. Researchers can investigate if a region exists where the real part of the ratio remains monotonic in σ for all t, potentially yielding a new sufficient condition for the Riemann Hypothesis.
q-Analogues and the Distribution of Primes: Investigating the q-zeta function and its ratio monotonicity. The q-parameter acts as a regulator, and analyzing how monotonicity is lost as q approaches 1 might provide refined estimates for zero-free regions.
Orthogonal Polynomials Associated with Zeta Ratios: Since completely monotonic functions are Laplace transforms of positive measures, there exists a sequence of orthogonal polynomials associated with the measure derived from ζ(x+1)/ζ(x). The asymptotics of these recurrence coefficients could be linked to the moments of the zeta function.

Computational Implementation

(* Section: Monotonicity Analysis of Zeta Ratios *)
(* Purpose: This code visualizes the ratio zeta(x+1)/zeta(x) and 
   numerically verifies the complete monotonicity property 
   discussed in arXiv:hal-04346907. *)

ClearAll[zetaRatio, dZetaRatio];

(* Define the primary ratio function *)
zetaRatio[x_] := Zeta[x + 1]/Zeta[x];

(* Define a function to compute the n-th derivative *)
dZetaRatio[x_, n_] := Derivative[n][zetaRatio][x];

(* 1. Plot the Zeta Ratio for x > 1 *)
plot1 = Plot[zetaRatio[x], {x, 1.1, 10}, 
  PlotStyle -> Thick, 
  AxesLabel -> {"x", "zeta(x+1)/zeta(x)"},
  PlotLabel -> "Ratio of Riemann Zeta Functions"];

(* 2. Verify the first derivatives alternate in sign *)
(* According to complete monotonicity, (-1)^n * f^(n)(x) >= 0 *)
derivTable = Table[
  {n, dZetaRatio[2.0, n], If[(-1)^n * dZetaRatio[2.0, n] >= 0, "True", "False"]},
  {n, 0, 5}
];

(* 3. Visualize the derivatives to show sign alternation *)
plot2 = LogPlot[
  Table[Abs[dZetaRatio[x, n]], {n, 1, 4}], 
  {x, 1.5, 5}, 
  PlotLegends -> {"f'(x)", "f''(x)", "f'''(x)", "f''''(x)"},
  PlotLabel -> "Magnitude of Derivatives (Log Scale)"];

Print["Monotonicity Check at x=2:"];
Print[TableForm[derivTable, TableHeadings -> {None, {"n", "f^(n)(2)", "CM Property"}}]];
GraphicsGrid[{{plot1, plot2}}]

Conclusions

The analysis in arXiv:hal-04346907 reinforces the idea that the zeta function possesses a profound level of regularity on the real line that is deeply connected to the non-negativity of the von Mangoldt function. This regularity serves as a necessary baseline against which the complex behavior of zeros must be measured. The most promising avenue for further research lies in the connection between these monotonicity properties and the Li criterion, potentially identifying a self-adjoint operator whose eigenvalues correspond to the zeros on the critical line.

References

Ismail, M. E. H., & Koelink, E. (2023). On a property of the Riemann zeta function. arXiv:hal-04346907
Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin.
Li, X.-J. (1997). The Positivity of a Sequence of Numbers and the Riemann Hypothesis. Journal of Number Theory, 65(2), 325-333.