The solution of Arnold’s problem on the weak asymptotics of Frobenius numbers with three arguments. (English. Russian original) Zbl 1255.11014

Sb. Math. 200, No. 4, 597-627 (2009); translation from Mat. Sb. 200, No. 4, 131-160 (2009).
The subject matter of the paper is the investigation of the asymptotic behavior of the Frobenius numbers (F.N.) with three arguments. It is shown that when summing over the set \[ M_a(x_1,x_2)=\left\{(b,c): 1\leq b\leq x_1a,\;1\leq c\leq x_2a,\;(a,b,c)=1\right\} \] F. N. behave like \(\frac{8}{\pi}\sqrt{abc}\). This result allows to prove Davison’s and Arnold’s conjectures concerning the mean behavior of F. N. The key ingredient of the proof is Rödseth’s formula for F. N. with three arguments. Rödseth’s formula gives a representation of F.N. in terms of the continued fraction of the special number. This allows to rewrite the mean value of F. N. as a sum of Rödseth’s function over the lattice and this sum can be replaced by the integral over the corresponding domain. The density can be expressed as the number of solutions of the system of linear equations. To obtain the desired estimates of the error term Kloosterman sums and van der Corput’s method are used.


11D07 The Frobenius problem
11D04 Linear Diophantine equations
11A55 Continued fractions
11D85 Representation problems
11L05 Gauss and Kloosterman sums; generalizations
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