Numberphile v. Math: the truth about 1+2+3+...=-1/12

TL;DR

The Numberphile video on 1+2+3+...=-1/12 contains several mistakes and misleading statements, but it is based on a genuine connection between the zeta function and the eta function. Analytic continuation and super summation are used to explain the connection.

Transcript

Welcome to the first Mathologer video of the year. Today it's about something very serious and so I'm wearing a totally black t-shirt. You all like Numberphile right? Me too, except for this one video over there in which they prove the infamous identity 1+2+3+...=-1/12 using some simple algebra that even kids in primary school should be able to fol... Read More

Key Insights

• 🎮 The Numberphile video contains several mistakes and misleading explanations.
• 🤘 The zeta function and the eta function are connected through analytic continuation.
• 👻 Analytic continuation allows for the extension of functions to larger domains.
• 🦸 Super summation is a method used to assign values to divergent series.
• ❎ The Riemann zeta function has interesting properties, such as its zeros and values at negative even integers.
• 🪈 The Ramanujan summation method captures certain aspects of naturally ordered series and is used in the theory of divergent series.

Questions & Answers

Q: What is the main claim made in the Numberphile video?

The video claims that 1+2+3+...=-1/12, which is incorrect.

Q: What methods are used to assign values to divergent series?

Analytic continuation and super summation are used to assign values to divergent series.

Q: What is the connection between the zeta function and the eta function?

The zeta function can be obtained from the eta function by replacing all the minuses with pluses. They are related through analytic continuation.

Q: Why does the Numberphile video contain mistakes and misleading statements?

The video oversimplifies complex mathematical concepts and uses incorrect calculations to explain the connection between 1+2+3+... and -1/12.

Summary & Key Takeaways

• The Numberphile video claims that 1+2+3+...=-1/12, but this is incorrect.

• The video utilizes incorrect calculations and misleading explanations.

• The connection between the zeta function and the eta function is explored.

• Analytic continuation and super summation are used to assign values to divergent series.