## Asymptotics of some number theoretic functions and an application to the growth of nilpotent groups |

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absolutely continuous affine transformation analogously approximate area of y,z assertion assume asymptotics beta function biggest minimum boundedness Bs(n comes from Lemma computation constant function contains exactly continuous function continuous step function convergence result Corollary 4.3 counted by T(n define distribution dual space ea,eb equality equivalent EULER totient function explicit finite following bounds follows from Lemma formula FOURIER transform functional equation Furthermore given graph hence integral KLOOSTERMAN sums lattice points Lemma B.l limit of step linear lower arm lowest terms main result minima minimum of g MOBIUS function mod q monotonically increasing nilpotent groups nonnegative notation number of lattice number theoretic estimates oooo lim PROOF Proposition 3.13 Proposition 5.2 proved rational numbers real numbers resp right continuous step saw-tooth function smooth functions suitable sum converges absolutely sum is dominated supremum Theorem 4.2 uniform limit uniformly unit interval xy(x