The bit on Wikipedia that follows the equation is part of the equation. It is the context that explains how to compute it. You need that bit because the equation uses extremely common syntax as short-hand for an operation that is different from the common one. If you write it in a way that doesn't rely on redefining common syntax like so "ζ(-1) = -1/12" it seems a lot less interesting doesn't it? Perhaps syntax confusion is the only interesting thing to it.
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The real problem is that the definition of the sum involving the limit of partial sums – limits that way too often "diverge" or "refuse to exist" – isn't the only definition or the best definition or the most natural definition that may be connected to the sum. There exist better definitions of the infinite sum – numerous definitions that turn out to be more natural in physics applications – and they generally produce the result −1/12−1/12. It is no trick or sleight-of-hand. The value −1/12−1/12 is really the right one and the rightness may be experimentally verified (using the Casimir effect).
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The ordinary sum of any collection of positive integers is not a fraction, let alone a negative one.
If we take the set { 1, 2, 3, ... 4 } the summation or partial summation of no subset of this set, finite or not, converges on any fraction or negative number. No matter what order we choose for traversing that set for generating a series, we never see anything resembling -1/12 as a partial sum or limit.
The ordinary arithmetic sum of any collection of positive integers is an integer which is strictly greater than each the integers.
The pages you're referencing are all crackpottery.
>> The pages you're referencing are all crackpottery.
So... I can trust Ramanujan and Abel, that published results on these things, and Terrence Tao that has a nice writeup, and a bunch of others, or I can trust HN user "Kazinator", who... published some middle-school algebra "proof" on HN.
Indeed, and "summation" is not the same as "addition".
Unfortunately, the syntax chosen for representing the sequence is that of an additive series, where a binary + operator is interposed between terms.
The semantics being shown does not seem to follow from a redefinition of that operator per se as stand-alone binary operator.
You really want to show this as, say
fun([1 2 3 4 ... ])
a function applied to a vector. Why the algebraic rules seem to work is because fun is a linear operator; i.e.
n fun([x0 x1 ... ]) = fun([nx0 nx1 ...])
and
fun(v0 + v1) = fun(v0) + fun(v1)
We can ply these rules back to the original 1 + 2 + 3 ... notation and then they look like algebra.
Basically, none of this means that the natural numbers add to -1/12; only that the sequence of natural numbers can be fed into some decimating calculation which ends up with -1/12.
Well, no kidding; the sequence of natural numbers can be fed into a decimating calculation which converges on any value you want, if you can freely choose the decimating calculation, and that calculation can be chosen to be linear operator.
which would be required to hold if this were addition. We can't change the positions of the terms. Why? Because they correspond to different powers in a power series.
