  The original Gelfond-Schnirelman method, proposed in 1936, uses polynomials
with integer coefficients and small norms on $[0,1]$ to give a Chebyshev-type
lower bound in prime number theory. We study a generalization of this method
for polynomials in many variables. Our main result is a lower bound for the
integral of Chebyshev's $\psi$-function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and connects the
subject to the potential theory with external fields generated by
polynomial-type weights. We also solve the corresponding potential theoretic
problem, by finding the extremal measure and its support.
