Yves-François Pétermann

About a theorem of Paolo Codecà's and Ω-estimates for arithmetical convolutions

Abstract A formula is proved, that can be used to obtain Ω-estimates in a certain class of arithmetical convolutions. It is applied to various functions, in particular to the classical error terms H(x) ≔ ∑ n ≤ x φ(n) n − 6x π 2 , G(x) ≔ ∑ n ≤ x n φ(n) − 315ζ(3)x 2π 4 + 1 2 log x , and F(x) ≔ ∑ n ≤ x σ(n) n − π 2 x 6 + 1 2 log x) , where φ, σ, and ζ are the Euler, divisor, and Riemann functions. The new estimates F(x) = Ω ± ( log log x) and G(x) = Ω ± ( log log x) are then obtained. As a further application we have N H (x) ≥ ( log log x) 3 2 ( log log log x) −2 where NH(x) denotes the number of changes in sign of the integer-valued function H(n). It is conjectured by Erdos that NH(x) ∼ cx.

About a theorem of Paolo Codecà's and Ω-estimates for arithmetical convolutions: Addendum

Abstract We obtain two sided Ω-estimates for the class of convolutions g(x) ≔ Σ n ≤ z α(n)n a f( x n ) , where f is a periodic function of period 1 satisfying a Kubert identity of order l > 1, where a ≤ −1, and where α(n) = 1 or the Moebius function μ(n). This applies in particular to the divisor functions Ga,k of Chowla and Walum, to the function P of Hardy and Littlewood, and to an error term related to the Euler phi function.