Distribution of Farey fractions with $k$-free denominators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908431174074368 |
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| author | Chahal, Bittu Chatterjee, Tapas Chaubey, Sneha |
| author_facet | Chahal, Bittu Chatterjee, Tapas Chaubey, Sneha |
| contents | We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.\] We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues of the classical results of Franel, Landau, and Niederreiter for $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$, particularly, deriving an equivalent form of the generalized Riemann hypothesis (GRH) for Dirichlet $L$-functions in terms of the distribution of $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$. Beyond examining the global distribution, we also study the local statistics of these sequences. We establish formulas for all levels ($k\ge 2$) of correlation measure. Specifically, we show the existence of the limiting pair ($k=2$) correlation function and provide an explicit expression for it. Our results are based upon the estimation of weighted Weyl sums and weighted lattice point counting in restricted domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution of Farey fractions with $k$-free denominators Chahal, Bittu Chatterjee, Tapas Chaubey, Sneha Number Theory 11B57 \sep 11J71 \sep 11K38 \sep 11L07 \sep 11L15 \sep 11M26 We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.\] We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues of the classical results of Franel, Landau, and Niederreiter for $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$, particularly, deriving an equivalent form of the generalized Riemann hypothesis (GRH) for Dirichlet $L$-functions in terms of the distribution of $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$. Beyond examining the global distribution, we also study the local statistics of these sequences. We establish formulas for all levels ($k\ge 2$) of correlation measure. Specifically, we show the existence of the limiting pair ($k=2$) correlation function and provide an explicit expression for it. Our results are based upon the estimation of weighted Weyl sums and weighted lattice point counting in restricted domains. |
| title | Distribution of Farey fractions with $k$-free denominators |
| topic | Number Theory 11B57 \sep 11J71 \sep 11K38 \sep 11L07 \sep 11L15 \sep 11M26 |
| url | https://arxiv.org/abs/2507.00228 |