How many degrees of freedom are associated with a test of whether a zero-order or multiple correlation equals zero?
The test of whether a single (zero-order Pearson) correlation equals zero is equivalent to a test that the standardised regression estimate equals zero. This, therefore, may be expressed as a t-test which, for a sample size of N, has N-2 degrees of freedom.
For testing the correlation between a single variable and more than one predictor we use the R-squared criterion which is produced in a linear regression. The test of whether R-squared is non-zero has a F distribution with p, N-p-1 degrees of freedom, for p predictors.