Explicit average orders: news and problems

Ramaré, Olivier

doi:10.4064/bc118-10

Cited by 4 publications

(2 citation statements)

References 30 publications

(36 reference statements)

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“…Ramaré and Akhilesh [8, Thm. 1.2] have given the constant 3.95 under the condition X ≥ 1, later improved by Ramaré himself [7] to 2.44 under the condition X > 1. From these last two bounds, it is not difficult to extend the range of estimation to X > 0, as we have done for example throughout Lemma 3.2, and these bounds continue to be better than the value 4.4.…”

Section: Consequencesmentioning

confidence: 99%

“…In [8] and [7], the critical exponent is reached by a completely different approach, using some results known as the covering remainder lemma and the unbalanced Dirichlet hyperbola formula as well as strong explicit bounds on some summatory functions involving the Möbius functions that oscillate, unlike in our case. Furthermore, it is important to point out that whereas a similar path to [8] or [7] could have been followed, the results there make use of specific properties of the functions being averaged and thus are not easy to generalize. This is the reason why [8, Thm.…”

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confidence: 99%

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Explicit averages of square-free supported functions: to the edge of the convolution method

Alterman

2022

Colloq. Math.

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We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular behavior on the prime numbers and observe how the nature of this method gives error estimations of order X −δ , where δ belongs to an open set I of positive reals. In order to have a better error estimation, a natural question is whether or not we can achieve an error term of critical order X −δ 0 , where δ0, the critical exponent, is the right endpoint of I. We answer this in the affirmative by presenting a new method that improves qualitatively almost all instances of the convolution method under some regularity conditions; now, the asymptotic estimation of averages of wellbehaved square-free supported arithmetic functions can be given with its critical exponent and a reasonable explicit error constant. We illustrate this new method by analyzing a particular average related to the work of Ramaré-Akhilesh (2017), which leads to notable improvements when imposing non-trivial coprimality conditions.

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Section: Consequencesmentioning

confidence: 99%

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confidence: 99%