On Moments of the Error Term of the Multivariable k-th Divisor Functions: Asymptotic Formulas and Upper Bounds

Core Concepts

This research paper investigates the behavior of error terms in the asymptotic formulas for multivariable k-th divisor functions, providing upper bounds for their moments and establishing an asymptotic formula for the mean square of the error term when k=3.

Abstract

• Bibliographic Information: Zhen Guo and Xin Li. (2024). ON MOMENTS OF THE ERROR TERM OF THE MULTIVARIABLE K-TH DIVISOR FUNCTIONS. arXiv preprint arXiv:2411.06656v1.

• Research Objective: This paper aims to analyze the error term in the asymptotic formula for the sum of the k-th divisor function over multiple variables, specifically focusing on deriving upper bounds for its moments and establishing an asymptotic formula for the mean square when k=3.

• Methodology: The authors utilize techniques from analytic number theory, including Dirichlet series, Perron's formula, and the Cauchy-Schwarz inequality, to analyze the error term. They decompose the error term into manageable components and leverage existing results on the mean square of the error term for the single-variable divisor function.

• Key Findings: The paper provides upper bounds for the mean square of the error term for k ≥ 4 and establishes an asymptotic formula for the mean square of the error term when k=3. Additionally, the authors derive an upper bound for the third power moment of the error term when k=3 and study its first power moment, leading to a result on its sign changes.

• Main Conclusions: The research demonstrates the utility of analytic number theory tools in understanding the behavior of error terms in asymptotic formulas for divisor functions. The derived bounds and asymptotic formulas contribute to a deeper understanding of the distribution of divisor functions.

• Significance: This work contributes to the field of analytic number theory by providing refined estimates and asymptotic formulas for error terms related to divisor functions, which are fundamental objects in number theory.

• Limitations and Future Research: The paper primarily focuses on upper bounds and specific cases for the asymptotic formulas. Further research could explore lower bounds for the error terms and investigate the possibility of extending the asymptotic formula for the mean square to cases where k > 3.

Stats

For k ⩾3, Voronoi proved that αk ⩽(k −1)/(k + 1) for k ⩾3.
Hardy showed that αk ⩾βk ⩾(k −1)/2k holds for k ⩾3.
Hardy and Littlewood proved that αk ⩽(k −1)/(k + 2) for k ⩾4.
Ivic provided a summary of results, including α3 ⩽43/96 and αk ⩽(3k −4)/4k (4 ⩽k ⩽8).
For k = 3, Tong developed a method for deriving an asymptotic formula for the mean square of the error term.
Heath-Brown proved an upper bound estimate for the third power moment of the error term when k=3.

Quotes

Key Insights Distilled From

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

by Zhen Guo, Xi... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06656.pdf

Deeper Inquiries

How can the techniques used in this paper be applied to study other arithmetic functions beyond divisor functions?

The paper focuses on analyzing the error term in asymptotic formulas for sums involving the multivariable k-th divisor function. The techniques employed, however, have broader applicability to other arithmetic functions beyond divisor functions. Here's how:

Multiplicativity: The core strength lies in exploiting the multiplicative nature of the divisor function. This property allows for expressing the Dirichlet series of the function as an Euler product, a crucial step in the analysis. Many other arithmetic functions, like Euler's totient function or the sum of squares function, are also multiplicative. The techniques of expressing their sums as multiple Dirichlet series, analyzing their poles and residues, and applying techniques like Perron's formula can be directly transferred.

Approximation by Smooth Functions: The paper utilizes smooth functions to approximate the error term. This is a standard technique in analytic number theory, and it can be applied to other arithmetic functions as well. The key is to find suitable smooth functions that closely approximate the behavior of the arithmetic function under consideration.

Mean Value Estimates: The authors derive mean square and higher-moment estimates for the error term. These estimates are crucial for understanding the average behavior of the error term. Similar mean value estimates can be sought for other arithmetic functions, providing insights into their oscillatory behavior.

Sign Changes: The study of sign changes of error terms, as done for ∆r,3(x) in the paper, can be extended. Understanding when an error term changes sign gives information about its oscillatory behavior, which is valuable for various applications.
In summary: The techniques of expressing sums as multiple Dirichlet series, utilizing properties like multiplicativity, approximating by smooth functions, deriving mean value estimates, and analyzing sign changes provide a robust framework applicable to a wide range of arithmetic functions beyond just divisor functions.

Could numerical computations provide insights into the sharpness of the derived upper bounds or suggest potential refinements?

Yes, numerical computations can be extremely valuable in this context. Here's how they can provide insights:

Testing the Sharpness of Bounds: The paper establishes upper bounds for various quantities related to the error term. Numerical computations can test the sharpness of these bounds by comparing them with actual computed values for a range of inputs. If the computed values consistently stay significantly below the theoretical upper bounds, it might suggest that the bounds are not tight and could potentially be improved.

Identifying Patterns and Conjectures: Numerical computations can reveal patterns and trends in the behavior of the error term that might not be immediately obvious from the theoretical analysis. These patterns can then lead to new conjectures and avenues for further theoretical investigation, potentially leading to refined bounds or even asymptotic formulas in some cases.

Estimating Constants: Often, theoretical bounds involve unspecified constants. Numerical computations can help estimate the size of these constants, providing a more concrete understanding of the bounds.
However, some caveats exist:

Computational Limitations: Computing arithmetic functions, especially for large inputs, can be computationally expensive. This limits the range of values that can be feasibly computed.
Numerical Errors: Numerical computations are inherently subject to rounding errors and other numerical instabilities. Care must be taken to minimize and control these errors to ensure the reliability of the results.
In conclusion: While numerical computations have limitations, they can be a powerful tool for gaining insights into the sharpness of theoretical bounds, suggesting potential refinements, and guiding further theoretical research on the behavior of error terms in asymptotic formulas for arithmetic functions.

What are the implications of understanding the error terms in these asymptotic formulas for practical applications in fields like cryptography or coding theory?

While the study of error terms in asymptotic formulas for arithmetic functions might appear purely theoretical, it has significant implications for practical applications, particularly in cryptography and coding theory:

Cryptography:

Distribution of Prime Numbers: The distribution of prime numbers is fundamental to many cryptographic algorithms, such as RSA. Understanding the error terms in asymptotic formulas related to prime-counting functions can provide insights into the distribution of primes within specific intervals, which is crucial for key generation and security analysis.

Lattice-Based Cryptography: Lattice-based cryptography relies on the hardness of certain lattice problems. The error terms in asymptotic formulas related to lattice quantities can impact the security analysis of these cryptosystems.

Cryptanalysis: Cryptanalysis often involves exploiting subtle biases or non-randomness in cryptographic primitives. Understanding the error terms in relevant asymptotic formulas can help cryptanalysts identify and exploit such biases, potentially leading to attacks.

Coding Theory:

Error-Correcting Codes: Error-correcting codes are essential for reliable data transmission and storage. The design and analysis of these codes often rely on properties of finite fields and related arithmetic functions. Understanding the error terms in asymptotic formulas for these functions can lead to improved code constructions and better understanding of their error-correction capabilities.

Complexity Analysis: The efficiency of coding and decoding algorithms is crucial. Asymptotic analysis, including the study of error terms, helps determine the computational complexity of these algorithms, guiding the design of practical and efficient coding schemes.
In essence: A deeper understanding of error terms in asymptotic formulas for arithmetic functions provides valuable tools for analyzing the security and efficiency of cryptographic and coding-theoretic constructions. This knowledge can lead to the development of more robust and efficient systems, as well as a better understanding of their limitations and potential vulnerabilities.