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On moments of the error term of the multivariable k-th divisor functions

Zhen Guo, Xin Li

Abstract

Suppose $k\geqslant3$ is an integer. Let $τ_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}τ_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+α_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<α_k<1$ are computable constants. In this paper we study the mean square of $Δ_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $Δ_{r,3}(x)$. We also get an upper bound for the third power moment of $Δ_{r,3}(x)$. Moreover, we study the first power moment of $Δ_{r,3}(x)$ and then give a result for the sign changes of it.

On moments of the error term of the multivariable k-th divisor functions

Abstract

Suppose is an integer. Let be the number of ways can be written as a product of fixed factors. For any fixed integer , we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}τ_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+α_k+\varepsilon}), \end{equation*} where and are computable constants. In this paper we study the mean square of and give upper bounds for and an asymptotic formula for the mean square of . We also get an upper bound for the third power moment of . Moreover, we study the first power moment of and then give a result for the sign changes of it.

Paper Structure

This paper contains 16 sections, 121 equations.

Table of Contents

  1. Introduction and Results
  2. Notation.
  3. Some lemmas
  4. Proof of Theorem \ref{['U B for M T']} and Expression of
  5. Proof of Theorem \ref{['mean square dayu4']}
  6. Mean Square of
  7. Proof of Theorem \ref{['mean square 3']}
  8. Mean Square of
  9. Mean Square of
  10. Evaluation of
  11. Estimates of and
  12. Other terms in (\ref{['int M_1 chaifen']})
  13. Conclusion
  14. Proof of Theorem \ref{['third moment']}
  15. Proof of Theorem \ref{['first moment']}
  16. ...and 1 more sections

Theorems & Definitions (8)

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