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On curious properties of the general size-biased distribution

Alexander E. Patkowski

TL;DR

The paper studies a general size-biased distribution associated with the density $v_k(x)$, obtained from a Mellin-type transform of the Riemann $\xi$-function, and proves $v_k(x)>0$ for $k\ge1$ along with a majorization $X_{k-1}$ majorizing $X_k$. It develops a functional-equation framework that yields an identity linking moments of $X_k$ to $\xi^k(y)$ and expresses a convolution relation $v_{m+k}(z)=\int_0^{\infty} v_k(x) v_m(xz) dx$. It connects to the Riemann hypothesis by showing RH is equivalent to the real-zero property of $\Xi_0(t)$, and extends to a family $\Xi_{\lambda,k}(t)$ that obeys a backward-heat equation, widening zero-distribution criteria. Overall, the work ties analytic number theory and probabilistic constructions to offer new insights into xi-function related transforms via size-biased random variables.

Abstract

We offer further results on a general size-biased distribution related to the Riemann xi-function we presented in [9] using the work of Ferrar. Curious properties associated with its expected value are presented, which are related to special functional equations. We also relate our observations to some recent developments related to the Riemann hypothesis.

On curious properties of the general size-biased distribution

TL;DR

The paper studies a general size-biased distribution associated with the density , obtained from a Mellin-type transform of the Riemann -function, and proves for along with a majorization majorizing . It develops a functional-equation framework that yields an identity linking moments of to and expresses a convolution relation . It connects to the Riemann hypothesis by showing RH is equivalent to the real-zero property of , and extends to a family that obeys a backward-heat equation, widening zero-distribution criteria. Overall, the work ties analytic number theory and probabilistic constructions to offer new insights into xi-function related transforms via size-biased random variables.

Abstract

We offer further results on a general size-biased distribution related to the Riemann xi-function we presented in [9] using the work of Ferrar. Curious properties associated with its expected value are presented, which are related to special functional equations. We also relate our observations to some recent developments related to the Riemann hypothesis.
Paper Structure (3 sections, 3 theorems, 27 equations)

This paper contains 3 sections, 3 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. A functional relation for the general size-biased distribution
  3. A relationship with the Riemann hypothesis

Key Result

Theorem 1.1

If $X_k$ is the general size-biased distribution corresponding to the density function $x^{-1}v_k(x),$ then for integers $m\ge1,$$z>0,$

Theorems & Definitions (6)

  • Theorem 1.1
  • proof
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof