Inequalities and Generalizations for the Summatory Function of the Sum of Digits in Integer Bases

Core Concepts

The paper revisits and generalizes inequalities for the summatory function of the sum of digits in a given integer base, proving that several known results can be deduced from a recent theorem on maximum mutational robustness in genotype-phenotype maps.

Abstract

The paper focuses on studying inequalities satisfied by the summatory function of the sum of digits in a given integer base. It starts by showing that the case p = 0 in Allaart's 2011 result can be deduced from Graham's 1970 inequality. The main contributions are: A variation on Theorem 5.1 from a 2023 paper by Mohanty et al. on maximum mutational robustness. Two generalizations of Theorem 5.1 from the 2023 paper, from which most known results in the literature can be deduced. A proof that Theorem 5.1 from the 2023 paper is optimal in a certain sense. Showing that Graham's inequality and its generalizations by Allaart and Cooper are consequences of the results in this paper. The paper also discusses potential generalizations of Allaart's 2011 result and raises several open questions for further research.

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Summing the sum of digits

by Jean-Paul Al... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2311.16806.pdf

Deeper Inquiries

Can Allaart's 2011 inequality be generalized by replacing the binary digit sequence with a more general sequence of positive real numbers

Allaart's 2011 inequality can potentially be generalized by replacing the binary digit sequence with a more general sequence of positive real numbers. However, for such a generalization to hold, certain conditions would need to be imposed on this sequence. One crucial condition could be that the sequence is non-decreasing but not excessively so, possibly ensuring that each term is not too far ahead of the previous one. This constraint could be formulated as ensuring that each term in the sequence is not significantly larger than the preceding term, possibly by a factor of a constant greater than or equal to 2. Additionally, the sequence may need to be bounded to prevent divergence or erratic behavior in the inequality.

What conditions would be needed on this sequence for such a generalization to hold

The concept of a "Graham-Allaart inequality" could potentially unify the results of Graham and Allaart. This inequality would aim to bridge the gap between Graham's theorem and Allaart's inequality, potentially encompassing both results within a broader framework. By establishing a unified inequality that incorporates the essence of both Graham's and Allaart's findings, researchers could streamline the understanding and application of these related concepts in the realm of digit sums and related mathematical areas.

Is there a "Graham-Allaart inequality" that could unify the results of Graham and Allaart

The inequalities discussed in the paper could indeed be approached through the use of (generalized) binomial coefficients. By leveraging the properties and relationships inherent in binomial coefficients, researchers may be able to derive new inequalities or extend existing ones in the context of digit sums and related mathematical structures. Exploring the connections between digit sums and binomial coefficients could lead to novel insights and potentially uncover deeper mathematical relationships that underpin these concepts.