The other day MathMatize posted
roses are red
books go on a shelf
1+2+3+4+ …
with a photo of Ramanujan on X.
This was an allusion to the bizarre equation
1 + 2 + 3 + … = − 1/12.
This comes up often enough that I wanted to write a post that I could share a link to next time I see it.
The equation is nonsense if interpreted in the usual way. The sum on the left diverges. You could say the sum is ∞ if by that you mean you can make the sum as large as you like by taking the partial sum out far enough.
Here’s how the equation is mean to be interpreted. The Riemann zeta function is defined as
for complex numbers s with positive real part, and defined for the rest of the complex plane by analytic continuation. The qualifiers matter. The infinite sum above does not define the zeta function for all numbers; it defines ζ(s) for numbers with real part greater than 1. The sum is valid for numbers like 7, or 42 −476i, or √2 + πi, but not for −1.
If the sum did define ζ(−1) then the sum would be 1 + 2 + 3 + …, but it doesn’t.
However, ζ(−1) is defined, and it equals −1/12.
What does it mean to define a function by analytic continuation? There is a theorem that essentially says there is only one way to extend an analytic function. It is possible to construct an analytic function that has the same values as ζ(s) when Re(s) > 1, where that function is defined for all s ≠ 1.
We could give that function a new name, say f(s). That is the function whose value at −1 equals − 1/12. But since there is only one possible analytic function f that overlaps with ζ(s) we go ahead and use the notation ζ(s) for this extended function.
To put it another way, the function ζ(s) is valid for all s ≠ 1, but the series representation for ζ(s) is not valid unless Re(s) > 1. There are other representations for ζ(s) for other regions of the complex plane, including for s = −1, and that’s what lets us compute ζ(−1) to find out that it equals −1/12.
So the rigorous but less sensational way to interpret the equation is to say
1−s + 2−s + 3−s + …
is a whimsical way of referring to the function defined by the series, when the series converges, and defined by its analytic continuation otherwise. So in addition to saying
1 + 2 + 3 + … = − 1/12
we could also say
1² + 2² + 3² + … = 0
and
1³ + 2³ + 3³ + … = 1/120.
You can make up your own equation for any value of s for which you can calculate ζ(s).