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arXiv · 2407.04584

On partial derivatives of some summatory functions

Gérald Tenenbaum
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Abstract

Let $f$ be a real arithmetic function and let $g:[1,\infty[\to{\mathbb R}$ be a smooth function. We describe two emblematic instances in which saddle-point estimates may be used to evaluate the frequency, on the set of integers $n\leqslant x$, of the event $\{f(n)\leqslant g(n)\}$ from those relevant to the event $\{f(n)\leqslant y\}$. The first example revisits Dickman's historical contribution to the theory of friable integers. The second is concerned with the distribution of the squarefree kernel of an integer.

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Gérald Tenenbaum. 2024-07-05. On partial derivatives of some summatory functions. https://arxiv.org/abs/2407.04584

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