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$M$-functions and screw functions originating from Goldbach's problem and zeros of the Riemann zeta function

Kohji Matsumoto, Masatoshi Suzuki

TL;DR

The paper develops an axiomatic framework for M-functions associated with oscillatory sums tied to the zeros of the Riemann zeta function, and connects these to screw functions and the distribution of secondary main terms in Goldbach-type problems. By introducing pairs \Pi=(\Omega,a) with precise summability and symmetry conditions, it derives explicit limit-distribution formulas for the corresponding amplitude sums and shows how the M-function encapsulates the value distribution, including support bounds. It further links screw-function properties to the Riemann hypothesis, showing that certain secondary-term functions become screw functions if and only if RH holds, and connects these to infinitely divisible distributions via characteristic functions and point masses at the origin. The paper also provides unconditional explicit formulas for H(X) and its generalizations H_\ell(X), both in terms of primes and zeros, and establishes the equivalence of alternative representations of secondary terms under RH, highlighting the deep structure linking analytic number theory, probability, and functional-analytic concepts in the study of Goldbach-type problems.

Abstract

We study the $M$-functions, which describe the limit theorem for the value-distributions of the secondary main terms in the asymptotic formulas for the summatory functions of the Goldbach counting function. One of the new aspects is a sufficient condition for the Riemann hypothesis provided by some formulas of the $M$-functions, which was a necessary condition in previous work. The other new aspect is the relation between the secondary main terms and the screw functions, which provides another necessary and sufficient condition for the Riemann hypothesis. We study such $M$-functions and screw functions in generalized settings by axiomatizing them.

$M$-functions and screw functions originating from Goldbach's problem and zeros of the Riemann zeta function

TL;DR

The paper develops an axiomatic framework for M-functions associated with oscillatory sums tied to the zeros of the Riemann zeta function, and connects these to screw functions and the distribution of secondary main terms in Goldbach-type problems. By introducing pairs \Pi=(\Omega,a) with precise summability and symmetry conditions, it derives explicit limit-distribution formulas for the corresponding amplitude sums and shows how the M-function encapsulates the value distribution, including support bounds. It further links screw-function properties to the Riemann hypothesis, showing that certain secondary-term functions become screw functions if and only if RH holds, and connects these to infinitely divisible distributions via characteristic functions and point masses at the origin. The paper also provides unconditional explicit formulas for H(X) and its generalizations H_\ell(X), both in terms of primes and zeros, and establishes the equivalence of alternative representations of secondary terms under RH, highlighting the deep structure linking analytic number theory, probability, and functional-analytic concepts in the study of Goldbach-type problems.

Abstract

We study the -functions, which describe the limit theorem for the value-distributions of the secondary main terms in the asymptotic formulas for the summatory functions of the Goldbach counting function. One of the new aspects is a sufficient condition for the Riemann hypothesis provided by some formulas of the -functions, which was a necessary condition in previous work. The other new aspect is the relation between the secondary main terms and the screw functions, which provides another necessary and sufficient condition for the Riemann hypothesis. We study such -functions and screw functions in generalized settings by axiomatizing them.
Paper Structure (10 sections, 22 theorems, 128 equations)

This paper contains 10 sections, 22 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. The axiomatic framework
  3. Generalization of Theorem \ref{['thm_1_1']}
  4. Proof of Theorem \ref{['thm_2_1']}
  5. Examples satisfying condition \ref{['eq_0903_2']}
  6. A condition for $\Omega$ to be a subset of $\mathbb{R}$
  7. A class of screw functions
  8. A point mass formula at the origin for infinitely divisible distributions
  9. Explicit formulas for $H(X)$ and $H_\ell(X)$
  10. On the equivalence of \ref{['f_102']} and \ref{['f_106']}

Key Result

Theorem 1.1

We assume the Riemann hypothesis and the linear independence over rationals for the set of positive imaginary parts of the nontrivial zeros of the Riemann zeta function. Then, there exists an explicitly constructible density function $M_H: \mathbb{R} \to \mathbb{R}_{\geq 0}$, for which holds for any test function $\Phi: \mathbb{R} \to \mathbb{C}$ which is locally Riemann integrable. The function

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Corollary 2.1
  • proof
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • ...and 24 more