**G. Lomadze**

## On the Number of Representations of Positive Integers by the Quadratic Form *x*_{1}^{2} + ... + *x*_{8}^{2} + 4*x*_{9}^{2}

**abstract:**

An explicit exact (non asymptotic) formula is derived for the number
of representations of positive integers by the quadratic form
$x_1^2+\cdots+x_8^2+4x_9^2$. The way by which this formula is derived,
gives us a possibility to develop a method of finding the so-called
Liouville type formulas for the number of representations
of positive integers by positive diagonal quadratic forms in nine variables
with integral coefficients