The quadratic reciprocity law and the elementary theta function

This note points out a new aspect of the well-known relationship between the subjects mentioned in the title. The following result and its generalization in totally real algebraic number fields is central to the discussion. Let denote the Legendre symbol for relatively prime numbers a and b ℇ ℤ and a substitution of the modular subgroup Γ0(4). Then, if γ>0 and b≡1 mod 2, with and According to (1), the Legendre symbol behaves somewhat like a modular function ﹙apart from the known behaviour under and ﹚. (1) follows (see below) from the functional equation with provided that Here we used and always will use the abbreviation and ℇδ means the absolutely least residue of δ mod 4. In the proof, Hecke [4] assumed γ>0 (see also Shimura [5]).