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Jacob's ladders, point of contact of the remained in the prime-number law with the Fermat-Wiles theorem and multiplicative puzzles on some sets of integrals

Jan Moser

TL;DR

The paper addresses how the prime-number remainder $P(t)$ and the Riemann zeta-function on the critical line interact under the Riemann hypothesis, by introducing Jacob's ladders and new $Pζ$-functionals that connect with Fermat-Wiles via $Pζ$-equivalents. The approach builds a framework of increments of Ingham's integral, derives asymptotic mappings from $-P(t)$ to the prime-counting function $\pi(t)$, and establishes factorization and multiplicative relations among zeta-square moments evaluated along Jacob's ladders. Key contributions include the first $Pζ$-functional, a RH-based asymptotic mapping $-P(t) \to π(t)$, several $Pζ$-equivalents of Fermat-Wiles, and a family of asymptotic multiplicative formulas that relate Ingham-type integrals to products of zeta-moments over ladder intervals. These results provide a novel $Pζ$-framework linking prime distribution, zeta moments, and deep number-theoretic conjectures with potential implications for understanding the interaction between primes and zeta-values.

Abstract

In this paper we prove, on the Riemann hypothesis, the existence of such increments of the Ingham integral (1932) that generate new functionals together with corresponding new $Pζ$-equivalents of the Fermat-Wiles theorem. We obtain also new results in this direction.

Jacob's ladders, point of contact of the remained in the prime-number law with the Fermat-Wiles theorem and multiplicative puzzles on some sets of integrals

TL;DR

The paper addresses how the prime-number remainder and the Riemann zeta-function on the critical line interact under the Riemann hypothesis, by introducing Jacob's ladders and new -functionals that connect with Fermat-Wiles via -equivalents. The approach builds a framework of increments of Ingham's integral, derives asymptotic mappings from to the prime-counting function , and establishes factorization and multiplicative relations among zeta-square moments evaluated along Jacob's ladders. Key contributions include the first -functional, a RH-based asymptotic mapping , several -equivalents of Fermat-Wiles, and a family of asymptotic multiplicative formulas that relate Ingham-type integrals to products of zeta-moments over ladder intervals. These results provide a novel -framework linking prime distribution, zeta moments, and deep number-theoretic conjectures with potential implications for understanding the interaction between primes and zeta-values.

Abstract

In this paper we prove, on the Riemann hypothesis, the existence of such increments of the Ingham integral (1932) that generate new functionals together with corresponding new -equivalents of the Fermat-Wiles theorem. We obtain also new results in this direction.
Paper Structure (6 sections, 88 equations)

This paper contains 6 sections, 88 equations.

Table of Contents

  1. Introduction
  2. Jacob's ladders: notions and basic geometrical properties
  3. The first $P\zeta$-functional and corresponding $P\zeta$-equivalent generated by the formula (\ref{['1.3']})
  4. Riemann hypothesis, transformation $\{-P(t)\}\to\pi(t)$ and new $P\zeta$-equivalent of the Fermat-Wiles theorem
  5. Some factorization formulas based on the Riemann hypothesis and the Ingham's integral
  6. The asymptotic multiplicative formula as multiplicative puzzle on a set of integrals

Theorems & Definitions (12)

  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • remark 6
  • remark 7
  • remark 8
  • remark 9
  • remark 10
  • ...and 2 more