A remark on the average number of divisors of a quadratic polynomial
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Archiv der Mathematik
A remark on the average number of divisors of a quadratic polynomial Dongxi Ye
Abstract. In recent work, we use Dudek’s method together with a result of Zagier to establish an asymptotic formula for the average number of divisors of an irreducible quadratic polynomial of the form x2 − bx + c with b, c integers. In this note, we remark that one can adopt the work of Hooley to derive a more precise asymptotic formula for the case x2 −bx+c with b2 − 4c not a square, and as a consequence, re-establish the weaker asymptotic formula given in our recent work by different arguments. Mathematics Subject Classification. 11N37, 11N56, 11D09. Keywords. Number of divisors, Quadratic polynomial, Zagier zeta function.
1. Introduction. During the past seven decades, due to its underlying applications in the Diophantine set problems, and connections to the fascinating quadratic number fields, the problem of determining the asymptotic behavior of the average number of divisors of a quadratic polynomial of the form x2 − bx + c with b, c integers, namely, σ0 (n2 − bn + c) as N → ∞, (1.1)
n≤N
where σk (n) := d|n dk is the sum-of-divisor function of weight k, has been drawing considerable attention from many mathematicians. In [2], Dudek proves that 6 σ0 (n2 − 1) = 2 N log2 N + O(N log N ), π n≤N
Dongxi Ye is supported by the Natural Science Foundation of China (Grant No. 11901586), the Natural Science Foundation of Guangdong Province (Grant No. 2019A1515011323) and the Sun Yat-sen University Research Grant for Youth Scholars (Grant No. 19lgpy244).
D. Ye
Arch. Math.
which concretely realizes Hooley’s prediction [5] that σ0 (n2 − r2 ) = A(r)N log2 N + O(N log N )
(1.2)
n≤N
for some constant A(r) for the case r = 1. Building upon Dudek’s work, in her recent work [6], Lapkova derives an asymptotic formula for the case x2 − bx + c reducible with distinct roots having the same parity, and remarkably shows that the fastest growing term is independent of b and c in such a case. As a consequence, Hooley’s prediction (1.2) is fully verified. Most recently, using binary additive divisor sums, Dudek, Pa´ nkowski, and Scharaschkin [3] improve Lapkova’s asymptotic formula, and give a more precise estimation for (1.1) for x2 − bx + c reducible. They explicitly find A1 (b, c) and A2 (b, c) for which 6 σ0 (n2 − bn + c) = 2 N log2 N + A1 (b, c)N log N π n≤N
+ A2 (b, c)N + O(N 2/3+ )
(1.3)
holds. On the other hand, for the case x2 − bx + c irreducible, things are getting ∞ more interesting as in such a case, values of the L-series L(s; χ) := n=1 χ(n) ns associated to a quadratic character χ at s = 1 are involved and always appear to be part of the asymptotic limit with respect to the fastest growing term. For example, for the case b = 0 and −c not a square, in [4], Hooley shows that σ0 (n2 + c) = 2gc (1)N log N + O(N ), n≤N
where gc (s) =
K(s) d L s; χ−c/d2 2s ζ(2s) 2 d d |c (d,2)=1
with
K(s) =
1+
1 2s
−1 ∞
ρc (2n ) , 2ns n=0
ρk (d) is defined by ρk (d) = |{0 ≤ x < d| x2 ≡ k
(mod d)}| =
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