1+2+3+4+...=-1/12？李永乐老师讲黎曼猜想（1）

Summary

In this video, Mr. Li Yongle delves into the Riemann hypothesis, a significant unsolved problem in mathematics, following Sir Atiyah's claim of a proof. The video aims to clarify misconceptions about the sum of natural numbers, exploring Euler's work on the zeta function and its convergence properties. It explains Euler's erroneous conclusion that the sum equals -1/12 and introduces the concept of analytic continuation, which Riemann used to extend the zeta function across the complex plane, correcting Euler's mistake and setting the stage for the Riemann hypothesis discussion in subsequent videos.

Takeaways

• 📢 Sir Atiyah, a renowned mathematician, claimed to have a proof for the Riemann hypothesis and planned to present it on September 24.
• 🧐 The Riemann hypothesis is a significant and unsolved mathematical conjecture that has persisted for over 150 years.
• 📚 The script introduces a series of three videos to explore the Riemann hypothesis due to its complexity.
• 🔢 It discusses the sum of all natural numbers and whether it equals -1/12, a question that has intrigued many.
• 👨‍🏫 Euler's work on the zeta function, ξ(s), is highlighted, which is foundational to understanding the Riemann hypothesis.
• ➡️ For s=1, the series ξ(s) represents the sum of the reciprocals of all natural numbers and is known to diverge (harmonic series).
• 📉 Euler found that for s>1, the series converges, and for s=2, it equals π²/6, offering a method to estimate the value of pi.
• 🤔 Euler's results for s<1, particularly s=-1, where he claimed the sum of all natural numbers equals -1/12, are controversial.
• 🚫 The script points out that Euler's method was flawed due to the misuse of the radius of convergence for the series.
• 🔍 The concept of 'analytic continuation' is introduced to explain the errors in Euler's approach and its importance in mathematics.
• 📈 The Riemann zeta function extends Euler's work to the complex plane and provides a unique continuation of the series beyond its original domain.
• 🚸 The script clarifies that the sum of all natural numbers is not considered to be -1/12 by mathematicians, dispelling a common misconception.

Q & A

What is the Riemann hypothesis?

The Riemann hypothesis is an important unsolved problem in mathematics that has remained unproven for over 150 years. It is related to the distribution of prime numbers and the behavior of the Riemann zeta function.

Who is Sir Atiyah and why is his announcement about the Riemann hypothesis significant?

Sir Atiyah is a renowned British mathematician who is a double winner of the Abel Prize and the Fields Medal. His announcement is significant because the Riemann hypothesis is considered one of the most challenging mathematical problems, and a proof would be a groundbreaking achievement.

What is the harmonic series and why does it diverge?

The harmonic series is the sum of the reciprocals of all natural numbers (1 + 1/2 + 1/3 + ...). It diverges because, although each term gets smaller, the sum increases without bound, as shown by grouping terms and comparing them to a series of 1/2's that add up to infinity.

What is the connection between Euler's work on the zeta function and the sum of all natural numbers?

Euler studied the zeta function, which is a series that, for certain values of s, can be related to the sum of natural numbers. Specifically, when s = -1, the zeta function is related to the sum of all natural numbers, although Euler's methods for evaluating it in this case were later found to be flawed.

What is the sum of the series 1 + 1/2^2 + 1/3^2 + ... according to Euler?

Euler found that the sum of this series, which is related to the zeta function when s = 2, is equal to π^2/6. This is one of the ways Euler contributed to the estimation of the value of pi.

Why were Euler's methods for evaluating the zeta function at s = -1 considered incorrect?

Euler's methods were incorrect because he used a series expansion that was only valid for s > 1 and then incorrectly applied it to s = -1. This led to a divergence issue, as the series on the right-hand side does not converge at s = -1.

What is analytic continuation and why is it important in understanding Euler's work on the zeta function?

Analytic continuation is a mathematical process that extends the domain of a function beyond its original convergence radius. It is important because it allows for the extension of Euler's series for the zeta function to complex numbers, which was not possible with the original series definition.

What is the Riemann zeta function and how does it extend Euler's work?

The Riemann zeta function is an analytic continuation of Euler's series for the zeta function, which allows it to be defined on the entire complex plane except for the point s = 1. It extends Euler's work by providing a well-defined function for complex numbers, not just real numbers greater than 1.

What is the correct value of the Riemann zeta function at s = -1, and why does it differ from Euler's conclusion?

The correct value of the Riemann zeta function at s = -1 is -1/12. This differs from Euler's conclusion because Euler's methods were flawed when applied to s = -1, but the result was coincidentally correct due to the properties of the analytic continuation of the zeta function.

Why is it incorrect to say that the sum of all natural numbers is -1/12 based on the Riemann zeta function?

It is incorrect because the sum of all natural numbers is not represented by the zeta function at s = -1 in the domain of Euler's original series. The value of -1/12 comes from the Riemann zeta function, which is an analytic continuation and not the same as the original series that diverges at s = -1.