86 RANDOM NUMBERS 3.3.3 not on c) have
86 RANDOM NUMBERS 3.3.3 not on c) have small partial quotients. In particular, the result of exercise 19 implies that the serial test on pairs will be satisfactorily passed if and only if a/m has no large partial quotients. The book Dedekind Sums by Hans Rademacher and Emil Grosswald (Math. Assoc. of America, Carus Monograph No. 16, 1972) discusses the history and properties of Dedekind sums and their generalizations. Further theoretical tests, including the serial test in higher dimensions, are discussed in Section 3.3.4. EXERCISES-First Set 1. [A4101 Express Z mod y in terms of the sawtooth and 6 functions. 2. [M20] Prove the replicative law, Eq. (10). 3. [HM%%?] What is the Fourier series expansion (in terms of sines and cosines) of the function f(s) = ((x))? b 4. [Ml91 If m = lOlo, what is the highest possible value of d (in the notation of Theorem P), given that the potency of the generator is lo? 5. [A4211 Carry out the derivation of Eq. (17). 6. [A4271 Let hh + kk = 1. (a) Show, without using Lemma B, that a@, k, c) = a(h, k, 0) + 12 for all integers c 2 0. (b) Show that if 0 < j < k, (c) Under the assumptions of Lemma B, prove Eq. (21). b 7. [A424] Give a proof of the reciprocity law (19), when c = 0, by using the general reciprocity law of exercise 1.2.4-45. b 8. [M.!?4] (L. Carlitz.) Let By generalizing the method of proof used in Lemma B, prove the following beautiful identity due to H. Rademacher: If each of p, Q, r is relatively prime to the other two, (The reciprocity law for Dedekind sums, with c = 0, is the special case r = 1.) 9. [A4401 Is there a simple proof of Rademacher s identity (exercise 
along the lines of the proof in exercise 7 of a special case?