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I mean, given the Mellin inverse integral $ \int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $, can we evaluate this integral, at least as $ x \rightarrow \infty $?

Can the same be made for $ \int_{c-i\infty}^{c+i\infty}dsF(s)\exp(st) $ as $ x \rightarrow \infty $?

Why or why not can this be evaluated in order to get the asymptotic behaviour of Mellin inverse transforms?

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yes we can evaluate above integral but it depends on F(s).what is your F(s).then we can see how to solve it.above integral is inverse mellin transform.Some times it is v difficult to find inverse,it all depends on what F(s) is.

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$F(s)= \frac{1}{\zeta(s)}$ , the problem here is the infinite number of poles of $ F(s) $ – Jose Garcia Apr 20 '12 at 9:28
@JoseGarcia I seriously doubt your going to find a simple asymptotic expression for the mertens function. – Ethan Feb 28 '13 at 9:12

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