On the Frobenius problem for three arguments (Q2821899)

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 \textit{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 \(a_1, a_2, \dots, a_d\). The \textit{genus} of a numerical semigroup \(S=\langle a_1,\ldots , a_d\rangle\) is the number \(n(S)=|\mathbb{N}\setminus S|.\)NEWLINENEWLINEIn the majority of problems related to Frobenius numbers, it is more convenient to consider the functions \( f(a_1,\ldots , a_d) = g(a_1,\ldots , a_d)+ a_1 + \cdots + a_d,\) and \(N(a_1,\ldots , a_d) = n(a_1,\ldots , a_d)+\frac{a_1}{2 } + \cdots + \frac{a_d}{2 }-\frac{1}{2 },\) which have the following multiplicative property: NEWLINE\[NEWLINE\begin{aligned} f(ha_1, \ldots, ha_{d-1},a_d)=h\cdot f(a_1, \ldots, a_{d-1},a_d),\\ N(ha_1, \ldots, ha_{d-1},a_d)=h\cdot N(a_1, \ldots, a_{d-1},a_d). \end{aligned}NEWLINE\]NEWLINENEWLINENEWLINEIt was proved by \textit{A. V. Ustinov} [Sb. Math. 200, No. 4, 597--627 (2009); translation from Mat. Sb. 200, No. 4, 131--160 (2009; Zbl 1255.11014)] that Frobenius numbers \(f(a,b,c)\) have weak asymptotic \(\frac{8}{ \pi}\sqrt{abc}\): NEWLINE\[NEWLINE \frac{1}{a^{3/2}|M_a(x_1,x_2)|} \sum\limits_{(b, c) \in M_a(x_1,x_2)} \left(f(a,b,c)-\frac{8}{\pi}\sqrt{abc}\right)=O(a^{-1/6+\varepsilon}).NEWLINE\]NEWLINE Here \(M_a(x_1,x_2)=\{(b,c):1\leq b\leq x_1a,1\leq c\leq x_2a,(a,b,c)=1\}\).NEWLINENEWLINEThe paper under review contains new results about average behaviour of another characteristics of numerical semigroup \(S(a,b,c)\). In particular the following theorem is proved: NEWLINE\[NEWLINE \frac{1}{a^{3/2}|M_a(x_1,x_2)|} \sum\limits_{(b, c) \in M_a(x_1,x_2)} \left(N(a,b,c)-\frac{64}{5\pi^2}\sqrt{abc}\right)=O(a^{-1/6+\varepsilon}).NEWLINE\]