In this answer by Carlo Beenakker he cites the following regularization formula: $$\int_0^\infty x^p\,dx\mathrel{"="}\frac{(-1)^{p+1}}{(p+1)(p+2)},\;\;p=0,1,2,\dotsc,$$ citing Tafazoli - Calculation of the vacuum energy density using zeta function regularization.

The formula cries manifestly wrong to me. In my opinion, the correct regularization formula is $$\operatorname{reg} \int_0^\infty x^p dx=\frac{B_{p+2}(1)-B_{p+2}(0)}{(p+1)(p+2)},$$ which essentially gives $0$ for positive $p$. Indeed, the zero value for regularization of integrals of monomials from zero to infinity can be verified with Mathematica, using the following code:

`f[x_] := x^p; Limit[s Sum[f[s x],{x,1,Infinity},Regularization->"Dirichlet"],s->0]`

`Out := ConditionalExpression[0, p > -1]`

This code uses Dirichlet regularization, which is essentially the same as Zeta regularization, which the linked paper claims to use.

Moreover, applying Fourier and Laplace transforms hint at the same regularization value, $0$.

That said, I wonder whether I missed something (like assumptions) in the linked paper? Is this result considered controversial at all?

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