Almost periodicity of the normalized representation numbers associated to positive definite ternary quadratic forms (Q1293128)

Let \(Q\) be a positive definite quadratic form over \(\mathbb{Z}\) in 3 variables and \(A(Q,n)\) the number of representations of the natural number \(n\) by \(Q\). The author proves, that the normalized numbers \(r_Q(n): =n^{-1/2} A(Q,n)\) are \(q\)-almost periodic \((q\geq 1)\) and calculates the Fourier coefficient. \textit{M. Kac} [Am. J. Math. 62, 122-126 (1940; Zbl 0023.00904)] proved this theorem in the case \(Q=x^2_1+ x^2_2+ \cdots+ x^2_d\) which \(d\geq 4\). The case \(d=3\) requires another proof, since here the singular series is not absolute convergent. Furthermore the author shows the existence of a very smooth limiting distribution for \(r_Q\) and he gives an asymptotic formula for the mean of \(A(Q,n)\) restricted to squarefree \(n\).