Sums of five, seven and nine squares
The paper studies the number of representations of positive integers \(n\) as sums of five, seven and nine squares. In the cases of five and seven squares formulas for the numbers of representations of \(n\) are reduced to those of square-free part of \(n\). A similar result is obtained in the case of the nine squares for some types of \(n\). Here it may be noticed that the sums of five or seven squares are examples of quadratic forms from one-class genera and it is known that in this case the number of representations may be expressed by the so-called singular series. The representation of the series as an infinite product over all primes and the formulas for all the multiples (even with more general assumptions and for any positive definite quadratic forms) were obtained by \textit{A. V. Malyshev} [Tr. Mat. Inst. Steklova 65, 1--212 (1962; Zbl 0135.09801)]. Later the result was reduced to the finite product and by this method numerous forms were discussed by various authors.
scientific article; zbMATH DE number 2093640