TY - JOUR
T1 - Correlation of arithmetic functions over Fq[T]
AU - Gorodetsky, Ofir
AU - Sawin, Will
N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/4/1
Y1 - 2020/4/1
N2 - For a fixed polynomial Δ, we study the number of polynomials f of degree n over Fq such that f and f+ Δ are both irreducible, an Fq[T] -analogue of the twin primes problem. In the large-q limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on Δ in a manner which is consistent with the Hardy–Littlewood Conjecture. We obtain a saving of q if we consider monic polynomials only and Δ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in Δ. This allows us to obtain additional saving from equidistribution results for L-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the Möbius function and divisor functions.
AB - For a fixed polynomial Δ, we study the number of polynomials f of degree n over Fq such that f and f+ Δ are both irreducible, an Fq[T] -analogue of the twin primes problem. In the large-q limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on Δ in a manner which is consistent with the Hardy–Littlewood Conjecture. We obtain a saving of q if we consider monic polynomials only and Δ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in Δ. This allows us to obtain additional saving from equidistribution results for L-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the Möbius function and divisor functions.
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U2 - 10.1007/s00208-019-01929-x
DO - 10.1007/s00208-019-01929-x
M3 - Article
AN - SCOPUS:85074814825
SN - 0025-5831
VL - 376
SP - 1059
EP - 1106
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 3-4
ER -