Having too much time on my hands …

A few months ago, there was a minor Internet Brouhaha over a video (hereafter referred to as the first video) which proports to show that
| (1) |

This was picked up by Phil Plait, the Bad Astronomer, and was followed up by a slightly more rigorous proof, https://www.youtube.com/watch?v=E-d9mgo8FGk ("the second video"), some backtracking/correcting, a Wikipedia entry, and a lot of web pages.

So, is equation (1) really true?

Not really, but kinda-sorta.

It is true that the sum

| (2) |

*approach*the result (1) by several different lines of attack.

The standard method, which is referred to in the second video quoted
above, starts with the Riemann Zeta Function:

| (3) |

This series only converges, as noted, when the real part of s is
greater than one, but it can be analytically continued (a process that I understand in principle but don't know how to put into practice here) to give you results for other values of s. In particular, it is possible to show (see equation 25.6.3) that

| (4) |

Now if you just set s = -1 in (3) it looks like you get equation (1). This isn't exactly true, but we can say that

| (5) |

→means

by a procedure whose description will not fit in the margin of this web page.

That's one way to do it. I'm going to go at it another way. It's akin to the procedure that is described in the first video, but, I hope, more rigourous, though I have probably missed a few steps. It may be equivalent to Ramanujan summation, but don't quote me on that.

| (6) |

| (7) |

*This*equation gives a finite value for all x, except the pole at x = 1, and is equal to (6) for |x| < 1. So we can justifiably say that (7) is the analytic continuation of (6). Also note that if we put x = −1 into (6) and evaluate it by using (7) we get the result

| (8) |

Now look at the derivative of (6) and (7). We'll call this S_{2}:

| (9) |

| (10) |

| (11) |

| (12) |

| (13) |

Now let's take a look at the difference between S_{2}(x) and
S_{2}(−x). Define

| (14) |

| (15) |

Now rearrange (14) to write

| (16) |

| (17) |

| (18) |

| (19) |

U_{2}(x) → 1 + 2 + 3 + 4 + … , x
→ 1^{-} (20)

Where the 1^{−} indicates that x → 1 from below. However, from (16) we have

| (21) |

*using this prescription*

| (22) |

So what's going on here? In the limit x → 1^{−} the n^{th} term in the sum within (17) tends to the value n. The trick is that for any value x < 1 there are only a finite number of c_{n}(x) which are positive, and *an infinite number that are negative*. Take a look at the graph below, which shows c_{n}(x) for several values of x:

Once we get x > 0.9999 or so, a significant number of the c_{n}(x) are close to 1. However, for any x < 1, there are an *infinite* number of c_{n}(x) which are negative. These have weights much smaller than the postive values, but there are a lot more of them. The net result is that the sum (17) is always negative, and tends to −1/12 as x → 1^{−}.

Is this a unique result? That is, do we always get

| (23) |

_{n}(x) which still have the property c

_{n}(x) → 1 as x → 1, place them in (17), and get something other than -1/12 as x →1. But I haven't investigated that, and I certainly couldn't prove a general case.

File translated with help from T

_{E}X by T

_{T}H, version 4.03.

On 1 Apr 2014, 18:17.

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