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Asymptotics of some number theoretic functions and an application to the growth of nilpotent groups. (English) Zbl 1112.11307
Bonner Mathematische Schriften 266. Bonn: Univ. Bonn. 67 S. (1994).
This is the author’s doctoral dissertation studying the asymptotic behavior of $$T(n)$$, the number of lattice points in the region $$R(n)=\bigcup_ {a+b+c=n}\{[-ab,ab]\times [-ac,ac]\}$$ where the union is taken over positive integers $$a$$, $$b$$, and $$c$$, with sum $$n$$. The author obtains the estimate $$T(n)=\frac 18n^ 4-Cn^ {8/3}+o(n^ {8/3})$$, where $$C$$ is a positive constant. The proof involves Fourier techniques, Hardy-Littlewood type dissection of the unit interval, and an estimate of Salié for Kloosterman sums.

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11P21 Lattice points in specified regions 20F18 Nilpotent groups 11L05 Gauss and Kloosterman sums; generalizations 11P55 Applications of the Hardy-Littlewood method