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The mean value of Frobenius numbers with three arguments. (English. Russian original) Zbl 1307.11036

Izv. Math. 76, No. 4, 760-819 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 4, 125-184 (2012).
Denote by \(\widehat{\mathbb{N}}^d\) the set of integer vectors in \(\mathbb{R}^d\) with positive coprime coefficients. For given \({\mathbf a} = (a_1,\ldots , a_d)\in\widehat{\mathbb{N}}^d\), the Frobenius number \(g({\mathbf a}) = g(a_1,\ldots , a_d)\) is defined as the largest integer which is not representable as a non-negative integer combination of \(a_1, a_2, \dots, a_d\). In the majority of problems related to Frobenius numbers, it is more convenient to consider the function \(f({\mathbf a}) = f(a_1,\ldots , a_d) = g(a_1,\ldots , a_d)+ a_1 + \cdots + a_d,\) which returns the largest integer which is not a positive integer combination of \(a_1, a_2, \dots, a_d\).
J. L. Davison conjectured, see [J. Number Theory 48, No. 3, 353–363 (1994; Zbl 0805.11025)], that the limit \[ \lim_{ N \to \infty } \frac{1}{\# ( \widehat{\mathbb{N}}^{ 3 } \cap [1,N]^3)} \sum\limits_{(a, b, c) \in \widehat{\mathbb{N}}^{ 3 } \cap [1,N]^3 } \frac{f(a,b,c)}{\sqrt{abc}} \] exists and is finite. The stronger form of this conjecture was proved by the reviewer [Sb. Math. 200, No. 4, 597–627 (2009); translation from Mat. Sb. 200, No. 4, 131–160 (2009; Zbl 1255.11014)]: \[ \frac{1}{\# ( \widehat{\mathbb{N}}^{ d } \cap N\times [1,N]^2)} \sum\limits_{(N, b, c) \in \widehat{\mathbb{N}}^{ 3 } \cap N\times [1,N]^2 } \frac{f(N,b,c)}{\sqrt{Nbc}}=\frac{8}{\pi}+O(N^{-1/12+\varepsilon}). \]
The paper under review devoted to original Davison conjecture. In particular it contains the following result: \[ \frac{1}{\# ( \widehat{\mathbb{N}}^{ 3 } \cap [1,N]^3)} \sum\limits_{(a, b, c) \in \widehat{\mathbb{N}}^{ 3 } \cap [1,N]^3 } \frac{f(a,b,c)}{\sqrt{abc}}=\frac{8}{\pi}+O(N^{-1/2+\varepsilon}). \]

MSC:

11D07 The Frobenius problem
11D45 Counting solutions of Diophantine equations
11L07 Estimates on exponential sums
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