Euler's Function on Products of Primes in Progressions

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Francis, Forrest
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Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Sciences
In 1962, Rosser and Schoenfeld asked whether there were infinitely many natural numbers n for which n/φ(n) > e^C log(log(n)), where C is the Euler-Mascheroni constant and φ(n) is Euler's totient function. This question was answered in the affirmative in 1983 by Jean-Louis Nicolas, who showed that there are infinitely many such n both in the case that the Riemann Hypothesis is true, and in the case that the Riemann Hypothesis is false. Landau's theorem naturally generalizes to the scenario where we restrict our attention to integers whose prime divisors all fall in a fixed arithmetic progression. In this thesis, I discuss the methods of Nicolas as they relate to the classical result, then turn my attention to answering the relevant analogue of Rosser and Schoenfeld's question for this restricted set of integers. It will be shown that, for certain arithmetic progressions, Nicolas' answer generalizes to a setting where the Generalized Riemann Hypothesis on a particular set of L-functions is concerned.
Research Subject Categories::MATHEMATICS , Euler-Mascheroni constant , Landau's theorem , Riemann hypothesis