Geodesic
A polynomial analogue of Jacobsthal function
Izvestiya. Mathematics, Tome 88 (2024) no. 2, pp. 225-235 Cet article a éte moissonné depuis la source Math-Net.Ru

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For a polynomial $f(x)\in \mathbb Z[x]$ we study an analogue of Jacobsthal function defined by $j_f(N) =\max_{m}\{$for some $x \in \mathbb N$ the inequality $(x+f(i),N)>1$ holds for all $i\leqslant m\}$. We prove the lower bound $$ j_f(P(y))\gg y(\ln y)^{\ell_f-1}\biggl(\frac{(\ln\ln y)^2}{\ln\ln\ln y}\biggr)^{h_f}\biggl(\frac{\ln y\ln\ln\ln y}{(\ln\ln y)^2}\biggr)^{M(f)}, $$ where $P(y)$ is the product of all primes $p$ below $y$, $\ell_f$ is the number of distinct linear factors of $f(x)$, $h_f$ is the number of distinct non-linear irreducible factors and $M(f)$ is the average size of the maximal preimage of a point under a map $f\colon \mathbb F_p\to \mathbb F_p$. The quantity $M(f)$ is computed in terms of certain Galois groups.
Keywords: Jacobsthal function, sieve
Mots-clés : polynomial, Galois group. 
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A. B. Kalmynin; S. V. Konyagin. A polynomial analogue of Jacobsthal function. Izvestiya. Mathematics, Tome 88 (2024) no. 2, pp. 225-235. http://geodesic.mathdoc.fr/item/IM2_2024_88_2_a2/

[1] K. Ford, B. Green, S. Konyagin, J. Maynard, and T. Tao, “Long gaps between primes”, J. Amer. Math. Soc., 31:1 (2018), 65–105 | DOI | MR | Zbl

[2] H. Iwaniec, “On the problem of Jacobsthal”, Demonstr. Math., 11:1 (1978), 225–231 | DOI | MR | Zbl

[3] R. A. Rankin, “The difference between consecutive prime numbers”, J. London Math. Soc., 13:4 (1938), 242–247 | DOI | MR | Zbl

[4] R. Dietmann, C. Elsholtz, A. Kalmynin, S. Konyagin, and J. Maynard, “Longer gaps between values of binary quadratic forms”, Int. Math. Res. Not. IMRN, 2023:12 (2023), 10313–10349 | DOI | MR | Zbl

[5] H. Halberstam and H.-E. Richert, Sieve methods, London Math. Soc. Monogr., 4, Academic Press, Inc., London–New York, 1974 | MR | Zbl

[6] J. C. Lagarias and A. M. Odlyzko, “Effective versions of the Chebotarev density theorem”, Algebraic number fields: L-functions and Galois properties (Univ. Durham, Durham, 1975), Academic Press, Inc., London–New York, 1977, 409–464 | MR | Zbl

[7] B. J. Birch and H. P. F. Swinnerton-Dyer, “Note on a problem of Chowla”, Acta Arith., 5 (1959), 417–423 | DOI | MR | Zbl

[8] J.-P. Serre, Topics in Galois theory, Res. Notes Math., 1, 2nd ed., A. K. Peters, Wellesley, MA, 2007 | MR | Zbl

[9] D. Hilbert, “Ueber die IrreduР±ibilität ganzer rationaler FunР±tionen mit ganzzahligen вЂ�oeffiР±ienten”, J. Reine Angew. Math., 1892:110 (1892), 104–129 | DOI | MR | Zbl