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February 6

Did Euler compute zeta(-n)?

I've been researching the history of the series 1 + 2 + 3 + 4 + ⋯, since it's currently a popular topic. I've seen lots of vague suggestions that Leonhard Euler wrote that 1 + 2 + 3 + 4 + ⋯ = −1/12 and 1 + 1 + 1 + 1 + ⋯ = −1/2, and it seems plausible that he did. After all, he wouldn't have written ζ(−1) = −1/12 or ζ(0) = −1/2, as we do today. The problem is that I've never actually found these claims in his writing, which is available at http://eulerarchive.maa.org. Granted, I've only looked in select documents, I don't read Latin, and it's possible that the claims are made in some subtle form that I missed.

So, did Euler really perform these computations? Can you prove it with a page number from a primary source?

Thanks! Melchoir (talk) 03:31, 6 February 2014 (UTC)[reply]

In chapter 2 of GH Hardy's Divergent Series, Hardy discusses how Euler used infinite series to "prove" a result equivalent to the complex functional equation. You might be able to find some of this on Google Books. Gutworth (talk) 04:07, 8 February 2014 (UTC)[reply]

I found the part I was refering to on Google Books. Gutworth (talk) 04:10, 8 February 2014 (UTC)[reply]

Thanks for the suggestion! The whole book is viewable at archive.org.

I've probably skimmed this chapter before, but reading it again strengthens my suspicion that Euler never computed these two quantities. Hardy writes that Euler stated the functional equation for the eta function, not the zeta function. Euler was more interested in summing oscillating series. The publication of Euler's being discussed, Remarques sur un beau rapport entre les series des puissances tant directes que reciproques (PDF), never even mentions a divergent series of positive terms.

Is that your conclusion as well? Melchoir (talk) 04:29, 8 February 2014 (UTC)[reply]

Yeah, I don't see

\zeta (-1)=-{\frac {1}{12}}

explicitly in there either. One can derive the

-{\frac {1}{12}}

result easily from the series he considers, so it's hard to imagine he didn't at some point compute it. There are a lot of Euler papers to go through... Gutworth (talk) 05:07, 8 February 2014 (UTC)[reply]

I searched for a while, but the only examples of divergent series of positive integers summing to a negative result in Euler's works that I found were of the geometric series type, like $1+2+4+8+\dots ={\frac {1}{1-2}}=-1$ , in De seriebus divergentis. While 1 + 1 + 1 + 1 + ⋯ and 1 + 2 + 3 + 4 + ⋯ are mentioned (without the result) as examples of various types of divergent series in the introduction, Euler's interest in that paper is in the series 1 − 1 + 2 − 6 + 24 − 120 + ⋯, which he deals with by taking differences. —Kusma (t·c) 19:44, 8 February 2014 (UTC)[reply]

Ah, right, 1 + 2 + 4 + 8 + ⋯ shows that Euler wasn't generally opposed to a series of positive terms having a negative result, and he talks about what that could mean. So close, and yet so far! Melchoir (talk) 21:11, 8 February 2014 (UTC)[reply]