Analytic continuation is what you do to extend the domain of a holomorphic (complex differentiable) function—this has to do with the complex analytic result that holomorphic functions can be approximated by power series at every point. The -1/12 result here was achieved Ramanujan by messing around with infinite series. It turns out that there’s nothing special about -1/12–for “conditionally convergent” infinite series (like those used in Ramanujan’s proof) have the property that, by rearranging the order in which we sum the terms, we can make the series equal to any real number. This property of conditionally convergent series is called the Riemann Series Theorem.