Sign changes of the Liouville function in arithmetic progressions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909013569961984 |
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| author | Ford, Kevin Radziwiłł, Maksym |
| author_facet | Ford, Kevin Radziwiłł, Maksym |
| contents | We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ such that $λ(m) = -1$ and $λ(n) = + 1$, where $λ$ denotes the Liouville function. Our result is motivated by Heath-Brown's explicit exponent in Linnik's theorem, establishing the existence of primes $p \equiv a \pmod{q}$ with $p \ll q^{5.5}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03349 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sign changes of the Liouville function in arithmetic progressions Ford, Kevin Radziwiłł, Maksym Number Theory We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ such that $λ(m) = -1$ and $λ(n) = + 1$, where $λ$ denotes the Liouville function. Our result is motivated by Heath-Brown's explicit exponent in Linnik's theorem, establishing the existence of primes $p \equiv a \pmod{q}$ with $p \ll q^{5.5}$. |
| title | Sign changes of the Liouville function in arithmetic progressions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.03349 |