A Note on Large Sums of Divisor-Bounded Multiplicative Functions
| Autor: | Frechette, Claire, Gerbelli-Gauthier, Mathilde, Hamieh, Alia, Tanabe, Naomi |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums associated to their products. More precisely, we let $f_j$, $j=1,2$ be such that $|f_j(n)|\leq \tau(n)^\kappa$ for some $\kappa\in\mathbb{N}$, and assume their partial sums satisfy $\left|S(x_j,f_j)\right|\geq \eta x_j (\log x_j)^{2^\kappa-1}$ for some $x_1, x_2\gg 1$ and $\eta>\max_j\{(\log x_j)^{-1/100}\}$. We then show that there exists $x\geq \min\{x_1, x_2\}^{\xi^2}$ such that $\left|S(x,f_1f_2)\right|\geq \xi x (\log x)^{2^{2\kappa}-1}$, where $\xi=C\eta^{1+2^{\kappa+3}}$ for some absolute constant $C>0$. Comment: 16 pages, project begun at Women in Numbers 6 |
| Databáze: | arXiv |
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