The cotangent identity on the lens space

This post is about a trigonometric sum, and I want to argue it deserves a post, because within it sits one of the cleanest corrections in this archive: a number that was bracketed correctly and added up wrongly, in the same line, by both a human and the literature it was copied from.

The setup

A lens space is a simple three-dimensional shape: a sphere with points identified by a rotation, here a rotation of order 6, written S cubed slash Z6. In the framework, the boundary geometry involves this quotient, and a quantity called the eta invariant measures the asymmetry of the spectrum of an operator on it. The eta invariant enters the vacuum-energy calculation, and its value on this shape is needed exactly, not approximately.

The APS theorem reduces the eta invariant to a finite sum over the rotation’s fixed points. For the order-6 quotient, the sum is: minus one sixth, times the sum over k from 1 to 5 of cotangent squared of pi k over 6.

The sum, by hand

The five cotangents squared. k equals 1: pi over 6 is 30 degrees, cot 30 is root 3, squared is 3. k equals 2: pi over 3 is 60 degrees, cot 60 is one over root 3, squared is one third. k equals 3: pi over 2 is 90 degrees, cot 90 is zero, squared is zero. k equals 4: two pi over 3 is 120 degrees, cot 120 is minus one over root 3, squared is one third. k equals 5: five pi over 6 is 150 degrees, cot 150 is minus root 3, squared is 3.

Sum: 3 plus a third plus 0 plus a third plus 3. That is 6 and two thirds, which is 20 over 3.

There is a classical identity for exactly this sum: the sum of cotangent squared from k equals 1 to p minus 1 of pi k over p equals p minus 1, times p minus 2, over 3. For p equals 6: 5 times 4 over 3, which is 20 over 3. The hand sum and the identity agree.

Now divide by minus 6: minus one sixth of 20 over 3 is minus 20 over 18, which is minus 10 over 9. That is the eta invariant: minus 10 over 9, about minus 1.111.

The error that lived in the literature

The published literature value for this eta invariant was minus 19 over 18. That corresponds to a sum of 19 over 3 rather than 20 over 3, which is to say one bracket of one third was added as 0, or one 3 was entered as 8 thirds. The error is a single slip in a five-term sum, and it propagated: the number was copied from paper to paper for years, because copying is easier than adding five numbers.

When the framework’s audit recomputed the sum, the first check was the classical identity, which is a theorem with several published proofs. 20 over 3. Then the direct evaluation, term by term. Also 20 over 3. Two routes, one answer, and the literature’s 19 over 3 was left with no route to it at all.

The corrected value matters downstream: the eta invariant is the topological part of the vacuum-energy calculation, and feeding minus 19 over 18 into it instead of minus 10 over 9 shifts a coefficient that other numbers lean on. This is exactly the kind of error that no amount of conceptual sophistication catches, because there is nothing conceptual about it. Only re-adding the sum catches it.

The discipline

The rule I take from this: any finite sum in a paper gets summed, by a second route, before it is used. The identity gives one route, direct evaluation gives another, and numerical evaluation gives a third. Here all three agree, and the agreement is the theorem. The embarrassing part is not that the literature had the slip. It is that for a while this programme quoted the literature value without doing the five additions, in a document whose entire method is re-adding sums. The correction is in the archive, the papers now carry minus 10 over 9, and the next post stays with this family of ideas to ask what an invariant like this actually buys.