#analyticcontinuation — Public Fediverse posts

@zvavybir
It does diverge. It has no sum.
However, the uniquely valued #Riemann #ZetaFunction can be analytically continued into the left half-plane where we find zeta(-1)=-1/12 (which 'looks like' 1+2+...). #Cesàro #summation will get you part of the way there also, and, as you say, yields the same result; presumably due to some ultimate cosmic logical rightness :-)
I very strongly recommend BP's superb exposition of this issue
https://www.youtube.com/watch?v=YuIIjLr6vUA
#maths #AnalyticContinuation #Ramanujan

For all \( a > 0 \) we have \[ \int_{-\infty}^\infty e^{-a x^2} = \sqrt{\frac{\pi}{a}} \]. Let's ignore the condition on \( a \) and say \( a \) is \( -1 \), then we have \[ \int_{-\infty}^\infty e^{x^2} = \sqrt{\pi} i \]. The left side obviously diverges and is never even slightly imaginary, but the right side is finite and purely imaginary. What is the connection between this integral and it's "value" (I'm pretty sure there is one, like how \( 1 + 2 + 4 + \dots = -1 \) makes sense over the 2-adic numbers)? Under what weird interpretation of integration is this correct? How can we find more of these weird integrals and their value?

#math #maths #mathematics #extendingMath #integration #analyticContinuation?

@Archimage

ζ(-1)=-1/12

#maths #AnalyticContinuation