Conway & Sloane (1988) give an expository account and precise statement of the mass formula for integral quadratic forms, which is reliable because they check it on a large number of explicit cases.
If f is an n-dimensional positive definite integral quadratic form (or lattice) then the mass of its genus is defined to be where the sum is over all integrally inequivalent forms in the same genus as f, and Aut(Λ) is the automorphism group of Λ.
The form of the mass formula given by Conway & Sloane (1988) states that for n ≥ 2 the mass is given by where mp(f) is the p-mass of f, given by for sufficiently large r, where ps is the highest power of p dividing the determinant of f. The number N(pr) is the number of n by n matrices X with coefficients that are integers mod p r such that where A is the Gram matrix of f, or in other words the order of the automorphism group of the form reduced mod p r. Some authors state the mass formula in terms of the p-adic density instead of the p-mass.
The factor of 2 in front represents the Tamagawa number of the special orthogonal group, which is only 1 in dimensions 0 and 1.
If all the p-masses have their standard value, then the total mass is the standard mass where The values of the Riemann zeta function for an even integers s are given in terms of Bernoulli numbers by So the mass of ƒ is given as a finite product of rational numbers as If the form f has a p-adic Jordan decomposition where q runs through powers of p and fq has determinant prime to p and dimension n(q), then the p-mass is given by Here n(II) is the sum of the dimensions of all Jordan components of type 2 and p = 2, and n(I,I) is the total number of pairs of adjacent constituents fq, f2q that are both of type I.
Then The functional equation for the L-series is where G is the Gauss sum If s is a positive integer then where Bs(x) is a Bernoulli polynomial.
For the case of even unimodular lattices Λ of dimension n > 0 divisible by 8 the mass formula is where Bk is a Bernoulli number.
Smith originally gave a nonconstructive proof of the existence of an even unimodular lattice of dimension 8 using the fact that the mass is non-zero.
The mass formula gives the total mass as There are two even unimodular lattices of dimension 16, one with root system E82 and automorphism group of order 2×6967296002 = 970864271032320000, and one with root system D16 and automorphism group of order 21516!
This implies that there are more than 80 million even unimodular lattices of dimension 32, as each has automorphism group of order at least 2 so contributes at most 1/2 to the mass.