Last time i thought i'd been careful enough, not to write

Nowadays it's only mildly surprising that an

The series

An oscillating series like

With that we can find an answer to the

...whose elements are just the factorial

Does that mean it's hard to come up with an method to arrive at any number? No, that's actually too easy. We're looking for "interesting" methods, ones that obey "interesting" rules. Like these:

A

The word

If you visualize the partial sums

The curve is some kind of smoothed version of our series. Initially defined only for integers, we can provide intermediate values by distorting our formula with the additional parmeter.

Smooth, like in

Meet

When that one

To learn about a related concept with prominence in physice, and often confused with zeta regularization, you may want to look up

Interestingly, all this connects Euler's

For nitpickers: Those aren't sums! You could still write them as sums or otherwise pretend they were, but you'd be missing a big chunk of beautiful mathematics...

Here's my post with the preceding discussion

https://plus.google.com/u/0/115434895453136495635/posts/Mux7Wctktvo

Here's a very nice, new and

http://wstein.org/rh/

https://archive.org/details/DivergentSeries

+John Baez has a nice post about -1/12 here:

https://plus.google.com/117663015413546257905/posts/j4Xxg44n1t6

Another post by +John Baez,

https://plus.google.com/u/0/117663015413546257905/posts/WoXqXCzkc9S

As always with him, don't miss the comments!

+Numberphile took it up as part of a series on Riemann's hypothesis. It attracted quite some fuss:

ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12

+The Aperiodical has collected a bulk of links of reactions to the above video:

http://aperiodical.com/2014/01/an-infinite-series-of-blog-posts-which-sums-to-minus-a-twelfth/

In that one's main post, a link to

http://www.slate.com/blogs/bad_astronomy/2014/01/17/infinite_series_when_the_sum_of_all_positive_integers_is_a_small_negative.html

#divergent #series #zeta #regularization #mathematics #sciencesunday

**1+2+3... = -1/12**directly, talking "to (pretend to) sum" it up. I wasn't. Now, after a nice and intense discussion with +Stam Nicolis (and a little reading) i'd like to share with you what came out of it.Nowadays it's only mildly surprising that an

*infinite sum*, or*series*can yield a*finite result*. Thanks to Baron**Augustin-Louis Cauchy**, we now teach clear criteria to students who are about the age he was, when he sanitized the mess that plagued algebraists before him. So that's the boring stuff: Made by a genius, simple to understand, highly practical.The series

**1+2+3...**on the other hand is very much ill behaved. There are other bad series which still behave better than this one. For example**+1-1+1-1...**where the*partial sums*oscillate between zero and one. We could just take the average and get**1/2**. While in such a case it seems reasonable to call the result a sum, it really is already something new:**1+2+3 ... n –> (1+2+3...n) / n**An oscillating series like

**+1-1+1-1...**can be handled by*Cesaro summation*, the fancy name for a family of methods similar to averaging. Instead of simply dividing by**n**one can imagine dividing by something larger, to capture faster growing, ill behaved series.**1+2+3 ... n –> (1+2+3...n) / e^(q·n)***Abel summation*amounts to dividing by an exponential function instead. Since that is a smooth function one can retain some smoothness by introducing an additional parameter, we can then again take a limit over. Roughly, let**q**go to zero like this:With that we can find an answer to the

*hypergeometric series*:**1-1+2-6+24-120...**...whose elements are just the factorial

**n!**peppered alternatingly with minus signs. For it, Euler derived the value**0.5963473623...**following half a dozen distinct ways! It's also interesting to note that*Abel summation*is*consistent*with*Cesar summation*, whenever the latter is defined, the former gives the same answer.Does that mean it's hard to come up with an method to arrive at any number? No, that's actually too easy. We're looking for "interesting" methods, ones that obey "interesting" rules. Like these:

**regular**: Yields the known result when applied to ordinary convergent series**linear**:**A(k·r + s) = k·A(r) + A(s)****stable**: Leaving off initial elements gives the same result.A

*regular*method is interesting because it can then be seen as a generalization of the common*Cauchy series*.*Linearity*is just a nice property also obeyed by them. As is*stability*. The*Abel sum*is all of these.The word

*sum*is already a bit of a stretch. Okay, classic*Cauchy series*are still used, and only the terms change: something gets multiplied to them. And what about that free parameter?If you visualize the partial sums

**1+2+3...**as a step function, you could put a parabola through the midpoints of the steps (look at the image now). A parabola feels right because the*partial sum*formula for our*triangle numbers*involves some**x^2**. Here's the thing: That parabola intersects the y-axis at**y = -1/12**!The curve is some kind of smoothed version of our series. Initially defined only for integers, we can provide intermediate values by distorting our formula with the additional parmeter.

Smooth, like in

*analytic continuation*! If we only could approach our series from another angle! So let's whack complext numbers through our formulas, they got an angle, eh? If it's smooth enough, we can extend our mutated series to the whole complex plane, and can then indeed approach our problematic value from another angle! It's called*Euler summation*, so he was there.Meet

*zeta regularization*, a nonlinear method that, of course, also gets us to**-1/12**:**1+2+3 ... n -> 1/1^s + 1/2^s + 1/3^s ...**When that one

*converges*for some**s**, and can from there be analytically continued to**s = -1**, the value obtained is called the*zeta regularized sum*of the original series.To learn about a related concept with prominence in physice, and often confused with zeta regularization, you may want to look up

*Dirichlet series*.Interestingly, all this connects Euler's

**-1/12**to the zeta function. He also liked to change the order of the elements in the series, and for that one may want to require**finite reindexability**: An alternative to*stability***Puzzle**: I'd like to know why i shouldn't confuse regularization with renormalization. The former i just told you about, the latter is required to keep the sums associated to*Feynman diagrams*from running off to infinity.For nitpickers: Those aren't sums! You could still write them as sums or otherwise pretend they were, but you'd be missing a big chunk of beautiful mathematics...

**references**Here's my post with the preceding discussion

*"Diagram 20: X-Rays of the zeta function"*https://plus.google.com/u/0/115434895453136495635/posts/Mux7Wctktvo

Here's a very nice, new and

**free**book by**Barry Mazur**,**William Stein**:*"Primes / What is the Riemann Hypothesis"*http://wstein.org/rh/

*"Divergent series"*, a classic book by**Godfrey Harold Hardy**, a friend of**Srinivasa Ramanujan**!https://archive.org/details/DivergentSeries

+John Baez has a nice post about -1/12 here:

https://plus.google.com/117663015413546257905/posts/j4Xxg44n1t6

Another post by +John Baez,

*"Prime numbers and the Riemann zeta function"*, a followup to +Matt McIrvin's post i linked last time:https://plus.google.com/u/0/117663015413546257905/posts/WoXqXCzkc9S

As always with him, don't miss the comments!

+Numberphile took it up as part of a series on Riemann's hypothesis. It attracted quite some fuss:

ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12

+The Aperiodical has collected a bulk of links of reactions to the above video:

http://aperiodical.com/2014/01/an-infinite-series-of-blog-posts-which-sums-to-minus-a-twelfth/

In that one's main post, a link to

**Phil Slate**, trying to put it on firm ground in a single blog entry:http://www.slate.com/blogs/bad_astronomy/2014/01/17/infinite_series_when_the_sum_of_all_positive_integers_is_a_small_negative.html

#divergent #series #zeta #regularization #mathematics #sciencesunday

View 9 previous comments

- No, it doesn't seem anybody did, +John Baez! Cool! Is it the paper version of your series on physics forums? I greatly enjoyed reading it, but failed to make the connection to this thread. My fault, I see what I should do now :)

Thanks!Apr 5, 2017 - The relevant section on Physics Forums is here:

physicsforums.com - Struggles with the Continuum - Part 5

I was trying to explain renormalization with an emphasis on what it really means, not on mathematical techniques (since that would require a textbook). So, I didn't bother to mention that the divergent integrals I discussed can often be made finite by replacing the dimension of spacetime (d = 4) by a variable d, and then figuring out what those integrals should equal when d is not an integer, getting an analytic function of d, which unfortunately has a pole at d = 4. This is called**dimensional regularization**, and it often involves zeta functions in practice.

But dimensional regularization is just one step in one particular approach to renormalization - and renormalization is much more exciting and profound, from a physics perspective, than this one step in one particular approach.

So, I explained a more old-fashioned, less mysterious approach. In both dimensional regularization and this old approach you replace the divergent integral you're interested in by something that converges but gives the*wrong answer*. In dimensional regularization you get a wrong answer depending on d, while in the approach I describe you get a wrong answer depending on the "momentum cutoff" Λ.

That's regularization. But then comes the interesting part: renormalization! Renormalization is the method of getting the*right*answer by taking the limit d → 4 or Λ → ∞*while changing various parameters as you do this!*

So, I tried to explain how this works and why it actually makes sense.Apr 5, 2017 - I just had the chance to refresh my memories with section 4 while attending parent's evening.

So renormalization is regularization on steroids, it begins with looking at integrals instead of sums. And from there builds a steampunk engine, with regularization gears, and cutoff pistons, to put the standard model in motion! Oh and one needs to wind it up backwards to make it end up right here...

Here's a reference that caught my eye, on the history of renormalization:**L.M. Brown**–*"Renormalization: From Lorentz to Landau (and Beyond)"*

Because now I'm even more curious how that could happen. Cool stuff, +John Baez, and right in time!Apr 5, 2017 - Oh wow. I just got entangled with
*dimensional regularization*. What the... that's... It doesn't even... Really?

https://en.wikipedia.org/wiki/Dimensional_regularization

Ah. Muahaha. Yeah, meromorphic nails are everywhere, if you're holding an analytic hammer.Apr 5, 2017 - No, you're right +Kanghun K! There's no way we can get a negative result from Abel summing positive numbers! I have no Idea why I would argue otherwise, and my post doesn't seem to claim such a thing. Thanks, however, for reviving this thread!Apr 5, 2017
- Ah! There's the exponential I was hallucinating:

en.wikipedia.org - Borel summation - Wikipedia

Thanks, +Kanghun K!Apr 26, 2017

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