Abstract
We show that the error term of the counting function of sums of two squares has oscillations of logarithmic frequency and size x1/2(log x)−3/2−ε. We do that by showing a theorem on oscillations for error terms of a rather general class of real functions. If the Mellin transform of f(x) has a singularity at ρ=β+iγ, γ≠0, with the principal summand of the form (s−ρ)−b(log(s−ρ))ch(s), h(ρ)≠0, we obtain oscillations of logarithmic frequency and size xβ(log x)b−1−ε. We also apply this result to other arithmetical functions related to sums of two squares.
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