Once introduced most symbols will remain effective throughout the sequel. Some of them are naturally standard. Thus ℤ, ℚ, ℝ, ℂ are sets of all integers, rationals, reals, and complex numbers, respectively. For example the group composed of all n × n integral matrices with determinant equal to 1 is denoted by SL(n,ℤ). The arithmetic functions σa(n) and dk(n) stand, respectively, for the sum of the ath powers of divisors of n and for the number of ways of expressing n as a product of k integral factors. In particular, d(n) = d2(n) is the divisor function. The Bessel functions are denoted by Iv, Jv, Kv as usual. We use the term K-Bessel function to indicate Kv without the specification of the order v; and the same convention applies to other Bessel functions as well. The symbol Г is for the gamma function, and Г is for the full modular group introduced in Section 1.1. The dependency of implied constants on others will not always be explained, since it is more or less clear from the context.

Some knowledge of integrals involving basic transcendental functions is certainly helpful. For this purpose Lebedev's book [38] is quite handy. But there are occasions when Titchmarsh [69], Watson [74], and Whittaker and Watson [75] give more precise information, though proofs of most integral formulas and relevant estimates are given or at least briefly indicated either in the text or in the respective notes.