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Explicit and Effective Estimates for the error term in the Generalised Divisor Problem

Neea Palojärvi, Sebastian Tudzi

Abstract

In this article, we obtain effective estimates for the error term $Δ_{k}(x)$ for all integers $k \geq2$, and completely explicit estimates for integers $k \in [3,9]$. The explicit results improve the powers of $x$ appearing in the known explicit bounds for $Δ_{k}(x)$, and the effective bounds provide a method to derive such bounds for all integers $k\geq 3$.

Explicit and Effective Estimates for the error term in the Generalised Divisor Problem

Abstract

In this article, we obtain effective estimates for the error term for all integers , and completely explicit estimates for integers . The explicit results improve the powers of appearing in the known explicit bounds for , and the effective bounds provide a method to derive such bounds for all integers .
Paper Structure (13 sections, 20 theorems, 221 equations, 4 tables)

This paper contains 13 sections, 20 theorems, 221 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Case distinction: Integer and non-integer $x$
  3. Decomposition and estimation of integrals
  4. Integrals in intervals $[b,c]$
  5. Applying moment and pointwise bounds on $\zeta(1/2+it)$
  6. Integral bound via the functional equation
  7. Effective results
  8. Cases where $x$ is not an integer or a half-integer
  9. Effective results based on the fourth moment
  10. Effective results based on the second moment
  11. An effective result based on the general estimate for an integral
  12. Proof of Theorem \ref{['thm:main']}
  13. Discussion

Key Result

Theorem 1.1

For $3\leq k \leq 9$ and $1\leq j \leq 3$, we have if $x_{k,j}\leq x<x_{k,j+1}$, where $j=1,2$, or $x \geq x_{k,3}$. Values $x_{k,j}$, $\alpha_{k,j}$, $\beta_{k,j}$ and $\gamma_{k,j}$ are presented in Table table:MainResults. Note that if $k\notin \{5,6\}$, then we only have the case $j=3$ meaning that we have only one estimate in all of the cases.

Theorems & Definitions (50)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 3.1
  • ...and 40 more