Geodesic
Asymmetrical problem about the lucky tickets
Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2015), pp. 19-26.

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We consider a set of tickets with numbers from $000001$ to $999999$. The lucky ticket is called, in which the sum of the first three digits equals the sum of the last three digits. The problem of lucky tickets is counting their number. The generalization of the classical problem of the lucky ticket to the case when the number of terms in comparable amounts and different radix arbitrarily. To count the number of solutions of linear diophantine equations, the terms of which are bounded by a constant, we introduced a discrete analogue of the delta function, written as the definite integral. We wrote out the formula for the number of tickets in the form of multiple sums comprising administering function. We prove several auxiliary identities for the end of trigonometric sums, it is a generalization of identities. After changing the order of summation and integration we obtain the desired formula for the number of solutions of this diophantine equation. With an equal number of terms in the sum written out expression obtained previously known classical result, which is known to the author sources obtained using more sophisticated techniques of integration in the complex domain. In this paper we use the same kind of method of trigonometric sums I.M. Vinogradov, and all the arguments are on the set of real numbers. To obtain an exact expression as an integral of the oscillating function method of stationary phase estimates are obtained, containing radicals and exponents. The results of the comparisons of calculations on the exact and approximate formulas are given.
Keywords: the method of trigonometric sums, method of stationary phase, happy tickets, asymptotic behavior, trigonometric identities. 
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I. G. Velichko. Asymmetrical problem about the lucky tickets. Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 3 (2015), pp. 19-26. http://geodesic.mathdoc.fr/item/VVGUM_2015_3_a2/

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[2] I.\;M. Vinogradov, The Method of Trigonometric Sums in Number Theory, Nauka Publ., M., 1971, 158 pp.

[3] I.\;S. Gradshteyn, Tables of Integrals, Series and Products, Fizmatgiz Publ., M., 1963, 1100 pp.

[4] M.\;V. Fedoruk, Asymptotics: Integrals and Series, Nauka Publ., M., 1971, 544 pp.