The sum 1+2+3+... = -1/12

What just happened?

This is one of the most surprising and most often misunderstood results in mathematics. The ordinary sum $1+2+3+4+\dots$ is of course divergent — the partial sums $S_N = 1+2+\dots+N = \tfrac{N(N+1)}{2}$ grow without bound. And yet, in a certain deeper sense, it is assigned the value $-\tfrac{1}{12}$.

Where does −1/12 come from?

From regularization using the Riemann zeta function. The function

is defined (convergent) only for $s>1$, but it can be analytically continued to other values of $s$. The point $\zeta(-1)$ formally corresponds to the “sum $1+2+3+\dots$” and equals:

Euler and Ramanujan

The result appeared in the works of Leonhard Euler, and was made famous by the renowned Indian mathematician Srinivasa Ramanujan (1913), who wrote it down in a letter to G. H. Hardy using his own method of summing divergent series.

A caution — what is really going on here?

This does NOT mean that adding the natural numbers gives a negative fraction. The addition $1+2+3+\dots$ remains divergent. $-\tfrac{1}{12}$ is an assignment of a meaningful, finite value to a divergent series by a specific method (regularization) — not the ordinary result of addition.

Where it shows up in physics:

The most astonishing thing is that this “strange” value $-\tfrac{1}{12}$ really appears in physics: in the Casimir effect (the force between two plates in the quantum vacuum) and in string theory (when determining the dimension of spacetime). It reveals a deep, non-obvious connection between the pure mathematics of divergent series and physics.

Remember: $1+2+3+\dots$ as an ordinary sum is divergent ($\to\infty$). The value $-\tfrac{1}{12}$ is the result of regularization with the zeta function ($\zeta(-1)=-\tfrac{1}{12}$) — the assignment of a finite number to a divergent series, not the effect of addition. Extension topic: a curiosity about the limits of the concept of a “sum”.