Proving ζ(-1)=-1/12 for Mathematicians

Context: Graduate
Thread starter moriheru
Start date Mar 17, 2015
Tags

Proof

Join the discussion

Registration is free. Ask a follow-up in this thread, or start your own.

2 replies · 1K views

moriheru

Thread starter
Mar 17, 2015

How can one prove the above equation?
Thanks.

Discussion

Mathematics news on Phys.org

Math shows why your vote may not count the way you think it does
Research cautions that sometimes working together can actually spur inequality
A new 'golden age' of mathematics may be dawning, thanks to AI and human ingenuity

micromass

Staff Emeritus

Science Advisor

Homework Helper

Insights Author

Mar 17, 2015

The hard part is actually defining the zeta function to the negative numbers. Once you have that, the equation follows easily.

A good proof can be found in following book by Godement: https://www.amazon.com/dp/3540209212/?tag=pfamazon01-20 Chapter VI.2.19

moriheru

Mar 17, 2015

Thanks micromass!

Proving ζ(-1)=-1/12 for Mathematicians

Discussion

Undergrad Is infinity a necessary concept?

High School Ant on a stretchy rope puzzle

High School Potato paradox

Undergrad Descartes’ Geometry of Square Roots

High School Three Squares Problem

Mathematica NDSolve with forcing function via "digital waveguide"

MATLAB Unable to produce desired results using state-space model with observer for inverted pendulum

Insights Revisiting the Velocity-Time Function

Insights Remote Operated Gate Control System

Insights AI Enriched Problem Solving

Insights Thinking Outside The Box Versus Knowing What’s In The Box

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

Proving ζ(-1)=-1/12 for Mathematicians

Discussion

Similar threads

High School What does the Ramanujan Summation of ζ(−1) = −1/12 represent precisely?

Graduate Is the sum of all natural numbers equal to -1/12?

Graduate How Can the Sum of All Natural Numbers Equal -1/12?

Undergrad Can the Infinite Sum of Natural Numbers Really Equal -1/12?

Undergrad What is up with people thinking that 1 + 2 + 3 + = -1/12?

Show Cos A = 1/12 in Square ABCD

Graduate Why does 1+2+3+4+... equal −1/12 if the series diverges?

Undergrad Moment of Inertia: Proving Mass, Side & Axis | 1/12 md²

Graduate Is the sum 1+2+3+4+... really equal to -1/12?

Solve Inequality: Algebraic Proof of a<b<c<1/12