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- W2317079023 abstract "Say that an integer n is y-ultrafriable if its canonical decomposition is free of prime powers exceeding y. We investigate the asymptotic behaviour of the distribution function ϒ(x, y), equal to the number of y-ultrafriable integers not exceeding x. The study being restricted to the range ψ(y)>2 log x (where ψ denotes Chebyshev's function) by a symmetry argument, and writing Ψ(x, y) for the number of y-friable integers not exceeding x, β=β(x, y) for the saddle point associated to the Dirichlet series |$Z(s,y):=prod_{pleqslant y}(1-p^{-(nu_{p}+1) s})/( 1-p^{-s})$| where |$nu_{p}:=lfloor(log y)/log prfloor$|, we obtain full description via a number of effective estimates, qualitative versions of which may be stated as follows: |$Upsilon(x, y)simPsi(x, y)Leftrightarrow y/(log x)^{2}to infty (xto infty)$|; |$Upsilon(x, y) = o(Psi(x, y))Leftrightarrow y/(log x)^{2}to 0 (xto infty)$|; |$Upsilon(x, y)sim x^{beta} Z(beta, y)G(beta sqrt{sigma_{2}}) (xto infty, 2 log x < psi(y)leqslant (log x)^{3})$|." @default.
- W2317079023 created "2016-06-24" @default.
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- W2317079023 date "2014-10-23" @default.
- W2317079023 modified "2023-10-14" @default.
- W2317079023 title "ON ULTRAFRIABLE INTEGERS" @default.
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- W2317079023 doi "https://doi.org/10.1093/qmath/hau025" @default.
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