3.3.3 THEORETICAL TESTS 87 10. [M%] (Web site domain) Show that
3.3.3 THEORETICAL TESTS 87 10. [M%] Show that when 0 < h < k it is possible to express a(k -h, Ic, c) and 0(h, k, -c) easily in terms of a(h, k, c). 11. [ARfO] The formulas given in the text show us how to evaluate c~(h, Ic, c) when h and Ic are relatively prime and c is an integer. For the general case, prove that a) ~(dh, dk, dc) = a(h, Ic, c), integer d > 0; b) a(h, k c + 0) = G, k c) + W cl~h integer c, real 0 < 0 < 1, when h and Ic are relatively prime and hh = 1 (modulo k). 12. [A&?,$] Show that if h is relatively prime to k and c is an integer, la(h, Ic, c)l 2 (k -l)(k -2)/k. 13. [M,z~] Generalize Eq. (26) so that it gives an expression for a(h, k, c). b 14. [MZO] The linear congruential generator that has m = 235, a = 2 + 1, c = 1, was given the serial correlation test on three batches of 1000 consecutive numbers, and the result was a very high correlation, between 0.2 and 0.3, in each case. What is the serial correlation of this generator, taken over all 235 numbers of the period? 15. [A&Z] Generalize Lemma B so that it applies to all real values of c, 0 5 c < k. 16. [MZd] Given the Euclidean tableau defined in (33), let po = 1, pl = al, and p, = a,p,-1 + p,-2 for 1 < j L t. Show that the complicated portion of the sum in Theorem D can be rewritten as follows, making it possible to avoid noninteger computations: c (-1)3+1& = & c (-q + b3(C3 + C,+1)P,-1. 3 3 ll?lt ll3lt [Hint: Prove that we have ~11j17(-1)3+1/m3m,+l = (-1)7f1p,-l/mlm,+tl for 1 5 r 2 t.1 17. [A&%?] Design an algorithm that evaluates cr(h, k, c) for integers h, k, c satisfying the hypotheses of Theorem D. Your algorithm should use only integer arithmetic (of unlimited precision), and it should produce the answer in the form A + B/k where A and B are integers. (Cf. exercise 16.) If possible, use only a finite number of variables for temporary storage, instead of maintaining arrays such as al, a2, . . . , at. b 18. [A&?31 (U. Dieter.) Given positive integers h, k, z, let Show that this sum can be evaluated in closed form, in terms of generalized Dedekind sums and the sawtooth function. [Hint: When z 5 k, the quantity lj/k] -L(j -z)/kJ equals 1 for 0 2 j < z, and it equals 0 for z 5 j < k, so we can introduce this factor and sum over 0 5 j < k.] b 19. [MZ3] Show that the serial test can be analyzed over the full period, in terms of generalized Dedekind sums, by finding a formula for the probability that a 2 X, < /3 and a 5 Xn+l < p when CY, p, CY , p are given integers with 0 5 cy < /3 2 m, 0 5 a < p 5 m. [Hint: Consider the quantity L(z -cu)/m] -L(z -P)/m].] 20. [MZ9] (U. Dieter.) Extend Theorem P by obtaining a formula for the probability that X, > Xntl > Xn+z, in terms of generalized Dedekind sums.
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