Creating Appealing SEO-Friendly Title: Euler Totient Function – Unveiling the Beauty of Monotone Non-Decreasing Sequences

Introduction:

I recently published a paper on the arXiv titled “Monotone non-decreasing sequences of the Euler totient function.” The paper explores the concept of the longest subsequence of numbers from 1 to n for which the Euler totient function is non-decreasing. I provide empirical evidence and conjectures regarding this sequence, as well as a new upper bound that improves upon previous results. The paper utilizes elementary methods, with the prime number theorem being the most advanced result used.

Full Article: Creating Appealing SEO-Friendly Title: Euler Totient Function – Unveiling the Beauty of Monotone Non-Decreasing Sequences

A Breakthrough in Number Theory: Monotone Non-Decreasing Sequences of the Euler Totient Function

In a groundbreaking paper titled “Monotone non-decreasing sequences of the Euler totient function,” a mathematician has just released a new research study on a fascinating topic in number theory. The paper delves into the concept of a quantity called θ, which is defined as the length of the longest subsequence of numbers from 1 to n for which the Euler totient function is non-decreasing.

Exploring the Behavior of the Euler Totient Function

The Euler totient function, denoted by φ(n), is a fundamental mathematical function that describes the number of positive integers less than n that are coprime with n. For example, φ(10) is equal to 4 because there are four numbers (1, 3, 7, and 9) less than 10 that are coprime with it.

The study initially investigates the values of θ and presents the first few values as A365339 in the OEIS (Online Encyclopedia of Integer Sequences). An interesting observation is made that the totient function is non-decreasing on the sets of primes and their powers, but not on other sets.

The Conjecture and Empirical Evidence

Based on extensive numerical evidence, a conjecture was proposed by Pollack, Pomerance, and Treviño. They believed that for all primes p, the maximum length θ would be achieved, meaning the totient function would be non-decreasing up to p. Although this conjecture has been verified up to a certain range, it remained unproved. The previous best upper bound was obtained by combining the results of previous studies, but it was not explicit.

The Asymptotic Solution

In this new research paper, the author presents an asymptotic solution for θ, which essentially determines its behavior as n approaches infinity. The result obtained answers a question posed by the renowned mathematician Paul Erdős and also addresses a closely related question by Pollack, Pomerance, and Treviño.

The Key Ideas and Methodology

The author reveals that the methods used in the proof of the asymptotic solution are mostly elementary. The only advanced concept required is the prime number theorem with a classical error term. The main idea behind the proof is to isolate a prime factor that has a significant influence on the behavior of the totient function φ(n).

By considering different factorizations of typical numbers, the author establishes an approximation that allows for a better understanding of when the totient function can be non-decreasing. This analysis, combined with the constraint of prime factorization, leads to the conclusion that θ must be piecewise constant for the totient function to be non-decreasing.

Achieving the Main Theorem

After carefully analyzing the exceptional cases where the previous approximation fails, the author successfully obtains the main theorem, provided a preliminary inequality holds for all positive rational numbers. The study highlights that this condition is not only necessary but also sufficient for the bound to hold.

Miracles and Counterexamples

In an interesting twist, the author discusses a minor miracle related to the largest prime factor of the denominator in the above inequality. This discovery allows for a precise evaluation of the left-hand side, facilitating the establishment of the inequality.

The author also brings attention to the fact that proving a similar result for the sum-of-divisors function would require a similar inequality, which currently lacks a complete proof in the literature. Additionally, the study explores counterexamples to a strong conjecture related to the totient function, shedding light on the challenges involved in proving it without making strong assumptions.

Looking Ahead

The research paper concludes with a discussion on near counterexamples to the conjecture, indicating the difficulty of proving it without assuming some additional hypotheses. The failure of Legendre’s conjecture and the Dickson-Hardy-Littlewood conjecture are presented as potential scenarios where the conjecture (related to monotone non-decreasing sequences of the Euler totient function) fails.

This research opens up new avenues of exploration in number theory and provides valuable insights into the behavior of the Euler totient function and its connection to prime numbers. The study’s elementary methodology makes it accessible to a wide range of mathematicians, and its impact on the field is expected to be significant.

Summary: Creating Appealing SEO-Friendly Title: Euler Totient Function – Unveiling the Beauty of Monotone Non-Decreasing Sequences

I have just uploaded my paper titled “Monotone non-decreasing sequences of the Euler totient function” to arXiv. The paper explores the longest subsequence of numbers from 1 to n for which the Euler totient function is non-decreasing. We provide an asymptotic proof and answer questions posed by Erdős, Pollack, Pomerance, and Treviño. Additionally, we discuss near counterexamples to the conjecture and the potential difficulties in proving it without strong assumptions.

Monotone Non-Decreasing Sequences of the Euler Totient Function – FAQs

Frequently Asked Questions

What is a monotone non-decreasing sequence?

A monotone non-decreasing sequence is a sequence in which each term is greater than or equal to the previous term. In other words, the sequence either stays the same or increases as you go from left to right.

What is the Euler totient function?

The Euler totient function, also known as Euler’s phi function, is an arithmetic function that counts the positive integers up to a given integer n that are relatively prime to n (i.e., the numbers that don’t share any common factors with n).

What does it mean to have a monotone non-decreasing sequence of the Euler totient function?

A monotone non-decreasing sequence of the Euler totient function means that each term in the sequence of values obtained by applying the Euler totient function to a sequence of integers is greater than or equal to the previous term. This implies that the values of the Euler totient function for successive integers are either the same or increase.

Why are monotone non-decreasing sequences of the Euler totient function important?

Monotone non-decreasing sequences of the Euler totient function have significance in number theory and related areas of mathematics. The behavior of these sequences provides insights into the distribution of prime numbers and can help in solving various mathematical problems.

Are all sequences of the Euler totient function monotone non-decreasing?

No, not all sequences of the Euler totient function are monotone non-decreasing. In fact, it is known that for any fixed positive integer, there exist sequences such that the values of the Euler totient function for successive integers can decrease as well. However, there are many interesting properties and patterns exhibited by monotone non-decreasing sequences.

Can you provide an example of a monotone non-decreasing sequence of the Euler totient function?

Sure! Let’s consider the sequence of integers from 1 to 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. When we apply the Euler totient function to each of these integers, we obtain the following sequence: 1, 1, 2, 2, 4, 2, 6, 4, 6, 4. As you can see, the values are either the same or increasing, hence forming a monotone non-decreasing sequence.

Additional Resources

For further information on monotone non-decreasing sequences and the Euler totient function, you may find the following resources helpful:

• [Link to Resource 1]
• [Link to Resource 2]
• [Link to Resource 3]