# §21.6 Products

## §21.6(i) Riemann Identity

Let be an arbitrary orthogonal matrix (that is, ) with rational elements. Also, let be an arbitrary matrix. Define

21.6.1

that is, is the set of all matrices that are obtained by premultiplying by any matrix with integer elements; two such matrices in are considered equivalent if their difference is a matrix with integer elements. Also, let

21.6.2

that is, is the number of elements in the set containing all -dimensional vectors obtained by multiplying on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. Then

where , , denote respectively the th columns of , , . This is the Riemann identity. On using theta functions with characteristics, it becomes

where and are arbitrary -dimensional vectors. Many identities involving products of theta functions can be established using these formulas.

## §21.6(ii) Addition Formulas

Let , , , . Then

Thus is a -dimensional vector whose entries are either 0 or 1. For this result and a generalization see Koizumi (1976) and Belokolos et al. (1994, pp. 38–41). For addition formulas for classical theta functions see §20.7(ii).