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An explicit result for short sums of positive arithmetic functions

Olivier Bordellès

TL;DR

The paper resolves the challenge of obtaining fully explicit, uniform bounds for short sums of nonnegative arithmetic functions obeying a mild growth condition, via a refined Erdős-Wolke–Shiu framework and Rankin-type methods. It proves a totally explicit Main Theorem bounding short sums in terms of $\sum_{n\le x} f(n)/n$ with explicitly defined constants, applicable to a broad class of functions satisfying $f(mn)\le \tau_k(m) f(n)$. The results yield explicit short-interval estimates for Hooley’s $Δ$-function, including a bound for $\sum_{x-y<n≤x} Δ(n)$ and a restricted-omega variant, both with computable constants. The combination of new Shiu refinements, explicit handling of non-multiplicativity via Rankin, and detailed prime-sum controls provides the first broadly explicit short-sum bounds in this setting, with direct consequences for the distribution of $Δ_k$ in short intervals. Overall, the work delivers fully explicit, practically usable bounds that enhance the quantitative understanding of short sums and Hooley’s Δ-function in analytic number theory.

Abstract

We prove a totally explicit bound for short sums of certain non-negative arithmetic functions satisfying a general growth condition, and apply this result to derive two explicit estimates for the Erdős-Hooley $Δ$-function in short intervals.

An explicit result for short sums of positive arithmetic functions

TL;DR

The paper resolves the challenge of obtaining fully explicit, uniform bounds for short sums of nonnegative arithmetic functions obeying a mild growth condition, via a refined Erdős-Wolke–Shiu framework and Rankin-type methods. It proves a totally explicit Main Theorem bounding short sums in terms of with explicitly defined constants, applicable to a broad class of functions satisfying . The results yield explicit short-interval estimates for Hooley’s -function, including a bound for and a restricted-omega variant, both with computable constants. The combination of new Shiu refinements, explicit handling of non-multiplicativity via Rankin, and detailed prime-sum controls provides the first broadly explicit short-sum bounds in this setting, with direct consequences for the distribution of in short intervals. Overall, the work delivers fully explicit, practically usable bounds that enhance the quantitative understanding of short sums and Hooley’s Δ-function in analytic number theory.

Abstract

We prove a totally explicit bound for short sums of certain non-negative arithmetic functions satisfying a general growth condition, and apply this result to derive two explicit estimates for the Erdős-Hooley -function in short intervals.
Paper Structure (22 sections, 22 theorems, 125 equations)

This paper contains 22 sections, 22 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main result
  4. Application
  5. Notation
  6. Proof of Theorem \ref{['th:main']}
  7. Tools
  8. Basics
  9. Key tools
  10. Proof of Theorem \ref{['th:main']}
  11. Class $\mathop{\mathrm{I}}\nolimits$
  12. Class $\mathop{\mathrm{II}}\nolimits$
  13. Class $\mathop{\mathrm{III}}\nolimits$
  14. Class $\mathop{\mathrm{IV}}\nolimits$
  15. Conclusion
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\ell \geqslant 1$ and $f$ be a positive arithmetic function such that there exists $k \in \mathbb {Z}_{\geqslant 1}$ such that, for all $(m,n) \in \left( \mathbb {Z}_{\geqslant 1} \right)^2$, we have Then, uniformly for all $x \geqslant \exp \left( 7^{28} (12e\ell)^{28\log ( 192 e \ell ) } \right)$ and $x^{1/ \ell} \leqslant y \leqslant x$, we have with $\Lambda (k,\ell) := 12 \ell k^{6 \el

Theorems & Definitions (38)

  • Theorem 1: Main Theorem
  • Corollary 2
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 28 more