This seminar has been cancelled.
The study on the mean values of the divisor function d(n)d(n) has a long history. Using his hyperbola method, Dirichlet established a well-known asymptotic formula for the mean values of d(n)d(n). P. Erdős established the correct size of the mean values of d(f(n))d(f(n)) for any irreducible f(x)∈ℤ[x]f(x)∈Z[x]. C. Hooley obtained asymptotic formulas for the mean values of d(n2+c)d(n2+c) when −c−c is not a perfect square. The mean values of divisor functions on binary quadratic forms were first studied by N. Gafurov and later by G. Yu, who proved asymptotic formulas for the mean values of d(n2+m2)d(n2+m2). In this talk, we show how to extend the work of Gafurov and Yu to establish asymptotic formulas for the mean values of d(n2+Nm2)d(n2+Nm2) for several NN. This is a joint work with L. Zhao.