Abstract
Reciprocity laws have held a special place in the theory of numbers ever since Gauss, at the age of nineteen, proved the first number-theoretic reciprocity law, the law of quadratic reciprocity. Since that time, many reciprocity laws have been discovered. Following a general discussion of reciprocity laws, we present a recently discovered reciprocity law for a well-known function of number theory, Ramanujan's sum [5]. The proofs of the reciprocity law are elementary and use concepts normally presented in an undergraduate number theory course. This is but another example demonstrating that once a fact is known, it is often not too difficult to prove.
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