On the Summation of Singular Series Associated with Certain Quadratic Forms (I)

The quadratic forms which are mainly dealt with in this paper are

a(x²+y²)+C(z² + t²), (1)

a(x² + y²)+2c(z² + t²), (2)

where x, y, z, t are integer variables and the constant coefficients a, c are unequal odd positive primes.