A Negative Number as Sum of All Natural Numbers

Marco Codegone ()
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Marco Codegone: Politecnico di Torino, Dipartimento di Scienze Matematiche

A chapter in Imagine Math 6, 2018, pp 115-123 from Springer

Abstract: Abstract In remarkable books in Theoretical Physics and String Theory an indication appears that the sum of the natural numbers is equal to minus one divided by twelve. Over the Internet one can find a proof which uses, in an arbitrary manner, computations with non-convergent series. The site has had a number of visitors well over one million. It would seem that mathematics is interesting when some results are presented that contrast common sense. It must be said that the study of not converging series has been developed by very important mathematicians from Euler to Ramanujan, giving us the opportunity to recall some important results on series and the analytic continuation. In this paper we intend to do so and to present the Ramanujan’s calculations highlighting the critical issues, recalling some theorems and indicating how minus one divided by twelve is linked to the analytic continuation of the Riemann zeta function.

Keywords: Ramanujan; Analytic Continuation; Important Mathematics; Divergent Series; Cesaro Summation (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-93949-0_10

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DOI: 10.1007/978-3-319-93949-0_10

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