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We establish an improved lower bound for a weighted counting function of representations of n as the sum of a prime and a square-free number and provide several applications for this bound. These applications are generalisations of a result of Dudek [4], which states that every natural number greater than 2 may be written as the sum of a prime and a square-free number.

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function ϕpnq and the Riemann Hypothesis. Among other things, we prove that for 1 ď q ď 10 and for q " 12, 14, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field Qpe 2πi{q q is true if and only if for all integers k ě 1 we have N k ϕpN k qplogpϕpqq log N k qq

Given a Dirichlet character χ modulo a prime q and its associated L-function, L(s, χ), we provide an explicit version of Burgess' estimate for |L(s, χ)|. We use partial summation to provide bounds along the vertical lines ℜs = 1 − r −1 , where r is a parameter associated with Burgess' character sum estimate. These bounds are then connected across the critical strip using the Phragmén-Lindelöf principle. In particular, for σ ∈ 1 2 , 9 10 , we establish|L (σ + it, χ)| ≤ (1.105)(0.692) σ q

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