Helfgott, Harald Andrés Square-free values of \(f(p)\), \(f\) cubic. (English) Zbl 1316.11084 Acta Math. 213, No. 1, 107-135 (2014). An integer is said to be square-free if it not divisible by the square \(d^2\) of any integer \(d\) greater than \(1\). The problem is considered on square-free numbers of the form \(f(p)\), \(p\) prime, in the case of a cubic polynomial \(f\). The following statement is the main result in the paper.Let \(f\in \mathbb{Z}[x]\) be a cubic polynomial without repeated roots. Then, the number of prime numbers \(p\leq n\) such that \(f(p)\) is square-free is \[ (1+o_f(1))\,\frac{n}{\log n}\prod\limits_{q \;\text{prime}}\left(1-\frac{\varrho_f(q^2)}{\varphi(q^2)}\right)+O(1). \] Here \(o_f(1)\) is a quantity that goes to zero as \(n\) tends to infinity at a rate that may depend on \(f\), \(O(1)\) is an absolute constant, \(\varphi\) is the Euler’s totient function and \(\varrho_f(q^2)\) is the number of solutions of congruence \(f(x)\equiv\text{{mod}}\,q^2\) in \((\mathbb{Z}/q^2\mathbb{Z})^*\). Reviewer: Jonas Šiaulys (Vilnius) Cited in 11 Documents MSC: 11N32 Primes represented by polynomials; other multiplicative structures of polynomial values 11N35 Sieves Keywords:square-free number; cubic polynomial; Galois group; elliptic curve; large sieve; large deviation estimate; multiset; enveloping sieve; Rankin-Selberg \(L\)-function × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Bombieri, E., Le grand crible dans la théorie analytique des nombres. Astérisque, 18 (1987). · Zbl 0618.10042 [2] Bombieri E., Friedlander J. B., Iwaniec H.: Primes in arithmetic progressions to large moduli. Acta Math. 156, 203-251 (1986) · Zbl 0588.10042 · doi:10.1007/BF02399204 [3] Breuil C., Conrad B., Diamond F., Taylor R: On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14, 843-939 (2001) · Zbl 0982.11033 · doi:10.1090/S0894-0347-01-00370-8 [4] Browning T. D.: Power-free values of polynomials. Arch. 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