Where do these infinites Tao is talking about come from?

Question

I was reading about how 1+2+3+4... !=-1/12 (which is something that drove me crazy when I first heard about it in a Numberphile video) in an article by Terence Tao. He says that -1/12 is in fact -1/12 + $ \int^{\infty}_{0} x \, dx $ . I have a couple of questions that I hope you guys can answer me in a way I could sort of understand as I'm still an undergraduate student. I've also read his article on zeta function but I didn't get a clue where do these integrals come from as it is a bit advanced for me.

My questions are: Where do these integrals come from? When he says "these infinites can sometimes be neglected as they can be considered orthogonal to the application" (near the end of the article), what does orthogonal to the application mean? How is the theory studies this infinite sums called? Any books where I can find more info?

Thanks in advance.

Answer 1

The construction comes from analytic continuation.

A function can only be considered valid where it is finite. Often if we extend its domain to complex numbers it remains finite for longer, indeed almost entirely.

A classic example is Riemann's extension of the zeta function.

$\zeta(-1)=-\frac{1}{12}$ with Riemann's zeta function, and $\infty$ with the basic zeta function.

The difference between the two values might be considered relevant, and could provide an insight as to how analytic continuation works.

Answer 2

Formulas like $$1+2+3+4+\ldots=-{1\over12}\tag{1}$$ are peddled even in the $21^{\rm st}$ century to impress simple minded people, and have per se no mathematical content whatsoever. It is true that one could replace the sum $\sum_{k=1}^\infty k$ on the left hand side of $(1)$ by a sum of the form $$\sum_{k=1}^\infty k\>\phi_k(z)$$ with certain suitably chosen functions $\phi_k(z)$ such that $(2)$ converges for some $z$, and then one could perform some limiting process $z\to\zeta$ whereby $\phi_k(z)\to 1$ for all $k\geq1$, and the sum in question converges to $-{1\over12}$. But none of this is exhibited in the crank formula $(1)$, and trumpeting about "analytical continuation" makes it no better. If all of this would not have the smell of RH nobody would seriously consider it.