Dedekind symbols with plus reciprocity laws (Q958651)

Certain generalizations of the classical Dedekind sums are studied in this paper. These generalizations are called Dedekind symbols and they are defined as complex valued functions on visible points in the right half plane. They are also assumed to satisfy the functional equation \(D(p,q)=D(p,p+q)\). Furthermore, if \(D(p,-q)=D(p,q)\) for all \((p,q)\) in the domain, then \(D\) is called an even Dedekind symbol. Odd Dedekind symbols are defined similarly. It turns out that \(D(p,q)\) can be uniquely determined up to additive constants by assuming a reciprocity law (similar to the reciprocity law for the classical Dedekind sums). There are two types of reciprocity laws that are postulated in this paper. Precisely, these are called plus or minus type laws. Many consequences of the less studied plus type reciprocity laws are given. These consequences indicate fundamental differences in the behavior of Dedekind symbols satisfying plus or minus type reciprocity laws.