Two objects are attached to a pair of polynomials f and g in p variables with positive coefficients: the Dirichlet series zeta(s;f,g) = sum over positive integer lattice points n of g(n) f(n)^{-s}, and the zeta integral Z(s;f,g) = integral over the positive octant of g(x) f(x)^{-s} dx. Both converge in a right half-plane and continue meromorphically to the whole complex plane. The central transfer rule says that the special value of the Dirichlet series at a non-positive integer s = -N is obtained from the corresponding integral by expanding the shifted integrand and replacing each monomial x_1^{a_1}...x_p^{a_p} by the product of Bernoulli numbers B_{a_1}...B_{a_p}, a substitution justified by Raabe's formula for Bernoulli polynomials. A student learns to compute zeta(-N;f,g) this way, to derive the degree-weighted product rule at s = 0 stating that zeta(0;f,g) for a product f = f_1 f_2 decomposes according to the degrees of the factors, and to recover Shintani's formula for cone zeta functions and Chen-Eie's evaluation as special cases. The same machinery settles Mahler's Hypothesis on the analytic behaviour of these series.
For polynomials f and g in p variables, the Dirichlet series zeta(s; f, g) sums g(k) f(k)^(-s) over all non-negative integer lattice points k in the octant, while the zeta integral Z(s; f, g) replace…