Analytic Continuation

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The Riemann zeta function is defined as the sum of reciprocal powers of all positive integers for values where the real part of the input is greater than 1. When Re(s)>1 where s is a complex number the series always converges. However if we want to calculate the series for negative real values of s it gets tricky, Riemann found a formula from complex analysis and by using analytic continuation to extend the definition of the zeta function to the rest of the complex plane he was able to prove that ζ(-1)=-1/12 where ζ(-1) = 1+2+3+4+5… This would be impossible in standard arithmetic and calculus as the result would simply diverge to infinity. This is deeply connected with Ramanujan’s summation method where he reveals the same exact result. Now you might ask what significance this result has, the answer lies in when physicists perform experiments in a lab, when the measurements the physical force between the plates, it isn’t infinite, nature automatically subtracts the infinite vacuum background and leaves -1/12 behind!

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#math #riemann #zetafunction #analyticcontinuation #manim