This volume is an outcome of

The 39th Taniguchi International Symposium on Mathematics

organized by myself, and of its forum, May 20–24, organized by N. Hirata-Kohno, L. Murata and myself as a conference at the Research Institute for Mathematical Sciences, Kyoto University.

I am deeply indebted to the Taniguchi Foundation for the generous support that made the symposium and the conference possible. The organizers of the conference acknowledge sincerely that the speakers were supported in part by the Inoue Science Foundation, the Kurata Foundation, Saneyoshi Scholarship Foundation, the Sumitomo Foundation; College of Science and Technology of Nihon University, the Research Institute for Mathematical Sciences of Kyoto University; and the Grant-Aid for General Scientific Research from the Ministry of Education, Science and Culture (through the courtesy of Prof. M. Koike, Kyushu University).

My special thanks are due to Profs. Hirata-Kohno and Murata for their unfailing collaboration during the three years of difficult preparation for the meetings.

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.

In this chapter we shall collect some basic facts about L-functions derived from cusp-forms over the full modular group Г by way of the Hecke correspondence. We shall see in the next chapter that these L-functions appear as components of our explicit formula for the fourth power moment of the Riemann zeta-function. This fact is in no sense superficial: In the course of the proof of the mean value we shall have an expression involving Kloosterman sums, which is spectrally decomposed by means of Theorems 2.3 and 2.5. But this will be made initially only in a domain of relevant parameters which does not contain the point we are most interested in. Thus we shall face the problem of analytic continuation; and its solution will depend indispensably on the analytical properties of these L-functions. In this process of analytic continuation the multiplicative property of Hecke operators will play an important rôle. Hence we shall develop the essentials of Hecke's idea in the first section. Once the analytic continuation is completed and the explicit formula for the mean value is established, we shall need good spectro statistical estimates of special values of L-functions to extract quantitative information from the formula. Having this application in mind we shall give an account of spectral mean values of Hecke L-functions in the later sections of this chapter.