A Note on Large Sums of Divisor-Bounded Multiplicative Functions
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| Format: | Preprint |
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2024
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| author | Frechette, Claire Gerbelli-Gauthier, Mathilde Hamieh, Alia Tanabe, Naomi |
| author_facet | Frechette, Claire Gerbelli-Gauthier, Mathilde Hamieh, Alia Tanabe, Naomi |
| contents | 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 $|f_j(n)|\leq τ(n)^κ$ for some $κ\in\mathbb{N}$, and assume their partial sums satisfy $\left|S(x_j,f_j)\right|\geq ηx_j (\log x_j)^{2^κ-1}$ for some $x_1, x_2\gg 1$ and $η>\max_j\{(\log x_j)^{-1/100}\}$. We then show that there exists $x\geq \min\{x_1, x_2\}^{ξ^2}$ such that $\left|S(x,f_1f_2)\right|\geq ξx (\log x)^{2^{2κ}-1}$, where $ξ=Cη^{1+2^{κ+3}}$ for some absolute constant $C>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_00658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Note on Large Sums of Divisor-Bounded Multiplicative Functions Frechette, Claire Gerbelli-Gauthier, Mathilde Hamieh, Alia Tanabe, Naomi Number Theory Primary 11F30, secondary 11F11, 11F12, 11M41 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 $|f_j(n)|\leq τ(n)^κ$ for some $κ\in\mathbb{N}$, and assume their partial sums satisfy $\left|S(x_j,f_j)\right|\geq ηx_j (\log x_j)^{2^κ-1}$ for some $x_1, x_2\gg 1$ and $η>\max_j\{(\log x_j)^{-1/100}\}$. We then show that there exists $x\geq \min\{x_1, x_2\}^{ξ^2}$ such that $\left|S(x,f_1f_2)\right|\geq ξx (\log x)^{2^{2κ}-1}$, where $ξ=Cη^{1+2^{κ+3}}$ for some absolute constant $C>0$. |
| title | A Note on Large Sums of Divisor-Bounded Multiplicative Functions |
| topic | Number Theory Primary 11F30, secondary 11F11, 11F12, 11M41 |
| url | https://arxiv.org/abs/2405.00658 |