insight - Scientific Computing - # Number Theory

On the Density of Natural Numbers Relatively Prime to the Floor of their Polynomial Values (For Certain Polynomial Classes)

Q: Can the results of this paper be extended to multivariate polynomials or other types of functions beyond polynomials?

Extending the results of this paper to multivariate polynomials or other functions is a natural and challenging direction for further research. Here's a breakdown of the potential challenges and considerations: Multivariate Polynomials: Increased Complexity of Diophantine Approximation: The core of the paper relies on controlling the behavior of Weyl sums, which in turn depends on the Diophantine properties of the coefficient of the linear term. In the multivariate case, this translates to analyzing simultaneous Diophantine approximations of multiple coefficients, significantly increasing the complexity. Geometric Considerations: With multivariate polynomials, the problem takes on a geometric flavor. Instead of points on a line, we're dealing with points on a surface or higher-dimensional variety. The distribution of these points modulo integers becomes more intricate. Potential for New Techniques: Addressing these challenges might require developing new tools and techniques in both analytic number theory (for handling the Weyl sums) and Diophantine geometry (for understanding the distribution of points on varieties). Beyond Polynomials: Loss of Algebraic Structure: Moving beyond polynomials to more general functions (e.g., analytic functions, exponential functions) means losing the convenient algebraic structure that polynomials provide. Techniques from algebraic number theory might no longer be directly applicable. Behavior of the Fractional Part: The paper heavily relies on understanding how the fractional part of ⌊P(n)⌋ behaves. For general functions, this behavior can be much more erratic and difficult to control. In summary: While extending the results to more general settings is an intriguing prospect, it poses significant hurdles. New ideas and approaches would likely be needed to overcome the increased complexity in Diophantine approximation and the loss of algebraic structure.

Q: What if we relax the condition on the coefficient of the linear term being non-Liouville? Are there cases where a different density might arise?

Relaxing the non-Liouville condition on the coefficient of the linear term is a crucial question that delves into the heart of the interplay between Diophantine approximation and the distribution of coprime pairs. Here's an exploration of the possibilities: Possibility of Different Densities: Liouville Numbers and Rational Approximations: If the coefficient of the linear term is a Liouville number, it admits very good rational approximations. This means that ⌊P(n)⌋ might be more likely to share factors with n, potentially leading to a lower density of coprime pairs. Resonance Phenomena: Certain irrational numbers, even if not Liouville, might exhibit "resonance" phenomena. This means that for specific sequences of integers, the fractional part of ⌊P(n)⌋ could become biased towards certain rational values, again potentially affecting the density of coprime pairs. Challenges in Analysis: Controlling Error Terms: The current proof relies heavily on the non-Liouville condition to control the error terms arising from Weyl sums. Relaxing this condition would necessitate finding alternative ways to bound these errors, which could be quite challenging. Case-by-Case Analysis: It's plausible that the density of coprime pairs might depend intricately on the specific Diophantine properties of the coefficient. A complete understanding might require a delicate case-by-case analysis. In essence: Relaxing the non-Liouville condition opens up a Pandora's box of possibilities. While it's conceivable that different densities might arise, rigorously proving such results would demand overcoming significant technical obstacles and potentially developing new tools in Diophantine approximation.

Q: How does the concept of coprimality in number theory relate to the idea of independence in probability and statistics, and are there deeper connections to explore?

The concept of coprimality in number theory and independence in probability, while seemingly distinct, share a fascinating underlying connection rooted in the idea of "lack of common structure." Exploring this connection can lead to fruitful insights: Coprimality as a Form of "Number-Theoretic Independence": Shared Factors as "Dependence": In number theory, two integers being coprime means they share no common factors other than 1. We can view this as a form of "independence" – the divisibility properties of one number provide no information about the divisibility properties of the other. Probability Analogy: Imagine choosing two integers at random. The probability that they share a common prime factor p is 1/p². As p gets larger, this probability decreases. Coprime integers, having no common prime factors, represent the extreme case of this "independence." Deeper Connections and Analogies: Chinese Remainder Theorem: This theorem, fundamental in number theory, essentially states that coprime moduli allow us to decompose congruences. This can be seen as analogous to how independence in probability allows us to decompose joint distributions. Probabilistic Number Theory: This branch of mathematics uses tools from probability to study the distribution of number-theoretic objects. The concept of coprimality plays a central role in many problems in this area. Ergodic Theory: This field explores the long-term average behavior of dynamical systems. There are intriguing connections between ergodic properties of certain systems and the distribution of coprime integers. In conclusion: While coprimality and independence arise in different mathematical contexts, they share a fundamental connection – the absence of shared structure. Exploring this connection through the lens of probability, ergodic theory, and other areas can lead to a deeper understanding of both number theory and the nature of randomness itself.

Core Concepts

This paper proves that for a polynomial with real coefficients where the coefficient of the linear term exhibits specific Diophantine properties (non-Liouville), the density of natural numbers relatively prime to the floor of their polynomial values converges to the inverse of the Riemann zeta function evaluated at 2 (i.e., 6/π²).

Abstract

Bibliographic Information: Chatterjee, A. (2024). On the Density of naturals n coprime to ⌊P(n)⌋ for certain Classes of Polynomials. arXiv:2411.11316v1 [math.NT].
Research Objective: To determine the density of natural numbers 'n' that are relatively prime to the floor of their corresponding polynomial values, ⌊P(n)⌋, for specific polynomial classes.
Methodology: The paper utilizes techniques from analytic number theory, including:
- Diophantine approximation to characterize the behavior of the polynomial's coefficients.
- Weyl's equidistribution theorem to analyze the distribution of fractional parts.
- Sieve theory to estimate the number of integers with desired coprimality properties.
Key Findings:
- The density of natural numbers 'n' relatively prime to ⌊P(n)⌋ is shown to be 1/ζ(2) (or 6/π²) when the coefficient of the linear term in the polynomial P(x) is non-Liouville.
- The proof relies on establishing bounds for specific Weyl sums and applying them in conjunction with the Erdos-Turan Discrepancy Theorem.
Main Conclusions: The research extends previous results on the density of coprime pairs (n, ⌊αn⌋) for irrational α to a broader class of functional relationships involving polynomial expressions.
Significance: This work contributes to understanding the distribution of coprime pairs in sequences generated by polynomial functions, with implications for number theory and related fields.
Limitations and Future Research: The current result focuses on polynomials with a non-Liouville coefficient for the linear term. Exploring the density for other polynomial classes or more general functions could be a direction for future research.

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Stats

The probability that two randomly chosen natural numbers are coprime is 6/π².
The density of natural numbers 'n' satisfying gcd(n, ⌊αn⌋) = 1, where α is irrational, is 6/π².

Quotes

"Roughly speaking, we show that if P is a polynomial with real coefficients and the coefficient of x in P is not too well-approximable, then the probability that n and ⌊P(n)⌋ are coprime is exactly 6/π²."
"This motivates us to proceed with sieving to establish the result rigorously. However, precise bounds require good control over the error terms, which arise when applying Weyl’s equidistribution theorem."

Key Insights Distilled From

On the Density of naturals $n$ coprime to $\lfloor P(n) \rfloor$ for certain Classes of Polynomials

by Aahan Chatte... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11316.pdf

$On the Density of naturals $n$ coprime to $\lfloor P(n) \rfloor$ for certain Classes of Polynomials$

Deeper Inquiries

Can the results of this paper be extended to multivariate polynomials or other types of functions beyond polynomials?

Extending the results of this paper to multivariate polynomials or other functions is a natural and challenging direction for further research. Here's a breakdown of the potential challenges and considerations:
Multivariate Polynomials:

Increased Complexity of Diophantine Approximation: The core of the paper relies on controlling the behavior of Weyl sums, which in turn depends on the Diophantine properties of the coefficient of the linear term. In the multivariate case, this translates to analyzing simultaneous Diophantine approximations of multiple coefficients, significantly increasing the complexity.
Geometric Considerations:  With multivariate polynomials, the problem takes on a geometric flavor. Instead of points on a line, we're dealing with points on a surface or higher-dimensional variety.  The distribution of these points modulo integers becomes more intricate.
Potential for New Techniques:  Addressing these challenges might require developing new tools and techniques in both analytic number theory (for handling the Weyl sums) and Diophantine geometry (for understanding the distribution of points on varieties).
Beyond Polynomials:

Loss of Algebraic Structure:  Moving beyond polynomials to more general functions (e.g., analytic functions, exponential functions) means losing the convenient algebraic structure that polynomials provide.  Techniques from algebraic number theory might no longer be directly applicable.
Behavior of the Fractional Part: The paper heavily relies on understanding how the fractional part of  ⌊P(n)⌋ behaves. For general functions, this behavior can be much more erratic and difficult to control.
In summary: While extending the results to more general settings is an intriguing prospect, it poses significant hurdles. New ideas and approaches would likely be needed to overcome the increased complexity in Diophantine approximation and the loss of algebraic structure.

What if we relax the condition on the coefficient of the linear term being non-Liouville? Are there cases where a different density might arise?

Relaxing the non-Liouville condition on the coefficient of the linear term is a crucial question that delves into the heart of the interplay between Diophantine approximation and the distribution of coprime pairs. Here's an exploration of the possibilities:
Possibility of Different Densities:

Liouville Numbers and Rational Approximations: If the coefficient of the linear term is a Liouville number, it admits very good rational approximations. This means that ⌊P(n)⌋ might be more likely to share factors with n, potentially leading to a lower density of coprime pairs.
Resonance Phenomena:  Certain irrational numbers, even if not Liouville, might exhibit "resonance" phenomena. This means that for specific sequences of integers, the fractional part of  ⌊P(n)⌋ could become biased towards certain rational values, again potentially affecting the density of coprime pairs.
Challenges in Analysis:

Controlling Error Terms: The current proof relies heavily on the non-Liouville condition to control the error terms arising from Weyl sums. Relaxing this condition would necessitate finding alternative ways to bound these errors, which could be quite challenging.
Case-by-Case Analysis: It's plausible that the density of coprime pairs might depend intricately on the specific Diophantine properties of the coefficient.  A complete understanding might require a delicate case-by-case analysis.
In essence: Relaxing the non-Liouville condition opens up a Pandora's box of possibilities. While it's conceivable that different densities might arise, rigorously proving such results would demand overcoming significant technical obstacles and potentially developing new tools in Diophantine approximation.

How does the concept of coprimality in number theory relate to the idea of independence in probability and statistics, and are there deeper connections to explore?

The concept of coprimality in number theory and independence in probability, while seemingly distinct, share a fascinating underlying connection rooted in the idea of "lack of common structure." Exploring this connection can lead to fruitful insights:
Coprimality as a Form of "Number-Theoretic Independence":

Shared Factors as "Dependence":  In number theory, two integers being coprime means they share no common factors other than 1.  We can view this as a form of "independence" – the divisibility properties of one number provide no information about the divisibility properties of the other.
Probability Analogy:  Imagine choosing two integers at random. The probability that they share a common prime factor p is 1/p².  As p gets larger, this probability decreases. Coprime integers, having no common prime factors, represent the extreme case of this "independence."
Deeper Connections and Analogies:

Chinese Remainder Theorem: This theorem, fundamental in number theory, essentially states that coprime moduli allow us to decompose congruences. This can be seen as analogous to how independence in probability allows us to decompose joint distributions.
Probabilistic Number Theory: This branch of mathematics uses tools from probability to study the distribution of number-theoretic objects. The concept of coprimality plays a central role in many problems in this area.
Ergodic Theory: This field explores the long-term average behavior of dynamical systems.  There are intriguing connections between ergodic properties of certain systems and the distribution of coprime integers.
In conclusion: While coprimality and independence arise in different mathematical contexts, they share a fundamental connection – the absence of shared structure. Exploring this connection through the lens of probability, ergodic theory, and other areas can lead to a deeper understanding of both number theory and the nature of randomness itself.