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8 votes

Series with the smallest number whose square is divisible by $n$

I couldn't find a reference, but (as noted in the OEIS page) if we have $k = a b^2$ with squarefree $a$ then $a(k) = ab$, so $$\begin{align*} \sum_{k\leq x}\frac1{a(k)} &= \sum_{a b^2 \leq x} \...
Command Master's user avatar
4 votes

Slick proof of Stirling's Formula?

I've played around with this a bit. I have a slick lower bound, but not a slick upper bound. We start with the $\Gamma$-integral: $$n! = \int_{x=0}^\infty x^n e^{-x} dx = \int_{y=-n}^\infty (n+y)^n e^{...
David E Speyer's user avatar
1 vote

History of asymptotic expansion of Laplace’s method between Laplace and Erdélyi

The article The origin of the method of steepest descent covers the pre-Debye history, tracing the method back to a 1829 paper by Cauchy, "Mémoire sur divers points d’analyse". Contributions ...
Carlo Beenakker's user avatar

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