Abstract
It is proved that as , uniformly for all positive integers , we have where. Here, is the Dickman function. We have and when , which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet L-functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for and . Furthermore, when assuming the Granville-Soundararajan conjecture is true, we establish the following asymptotic formulas: where is prime and isgiven.
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