Linnik's problem for multiplicative functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910264279957504 |
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| author | Matomäki, Kaisa Teräväinen, Joni |
| author_facet | Matomäki, Kaisa Teräväinen, Joni |
| contents | We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue class. We ask how far one must go before finding square-free integers $n_1,n_2\equiv a \pmod q$ with $h(n_1)<0<h(n_2)$. We show that one can always find such integers with $n_1,n_2\le q^{2+o(1)}$, unless the sign of $h$ strongly pretends to be a real Dirichlet character modulo $q$. Thus, apart from this natural character obstruction, sign changes of a multiplicative function occur in every reduced residue class at a scale corresponding essentially to the square root barrier. In the special case of the Liouville function $λ$ this improves on a recent result of Ford and Radziwiłł and matches, up to $q^{o(1)}$ factors, what was previously known conditionally under the generalized Riemann hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27833 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linnik's problem for multiplicative functions Matomäki, Kaisa Teräväinen, Joni Number Theory 11N13, 11B13 We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue class. We ask how far one must go before finding square-free integers $n_1,n_2\equiv a \pmod q$ with $h(n_1)<0<h(n_2)$. We show that one can always find such integers with $n_1,n_2\le q^{2+o(1)}$, unless the sign of $h$ strongly pretends to be a real Dirichlet character modulo $q$. Thus, apart from this natural character obstruction, sign changes of a multiplicative function occur in every reduced residue class at a scale corresponding essentially to the square root barrier. In the special case of the Liouville function $λ$ this improves on a recent result of Ford and Radziwiłł and matches, up to $q^{o(1)}$ factors, what was previously known conditionally under the generalized Riemann hypothesis. |
| title | Linnik's problem for multiplicative functions |
| topic | Number Theory 11N13, 11B13 |
| url | https://arxiv.org/abs/2605.27833 |