Number Theory and Its Applications II

The following sections are included:

• Preface
• Contents

The purpose of this chapter is to abridge a big gap between elementary number theory and algebraic number theory with the emphasis on similarity of the material with that of linear algebra. This is also a prelude to the theory of Hilbert spaces in Chapter 5. In elementary number theory what is essential is the unique factorization property of the ring (or rather an integral domain) 𝕫 of (rational) integers. Algebraically speaking, this is because 𝕫 is first of all a Euclidean domain. Then there is a theorem (Theorem 1.5, I) to the effect that a Euclidean domain is a PID (Principal Ideal Domain) in which all the ideals are principal. Another theorem (Exercise 16, I) asserts that a PID is a UFD (Unique Factorization Domain) in which every element is expressed as a product of irreducible elements up to units and order of irreducible elements. One often refers to irreducible elements as prime elements since they are the same in UFD. Since 𝕫 is an integral domain, it has the quotient field S⁻¹𝕫, where S = 𝕫\{0} is a multiplicatively closed subset. It is the rational number field ℚ of characteristic 0.

In this chapter we state group-theoretic aspects of algebraic numbers, which together with the linear-algebraic approach in Chapter 1 hopefully gives an easily accessible introduction to algebraic number theory. By this we present basic results without proof and we illustrate them by starting with the structure theorem on f.g. Abelian group applied to establishing the Dirichlet unit theorem, and we state rudiments of Galois theory since this is the main role played by group theory. We state some facts about cyclotomic fields since they are a good example of the theory and could work for an introduction to class field theory.

This chapter starts by developing elementary analytic number theory, with emphasis on the use of Stieltjes integrals, thereby stating three generating functions for arithmetic functions: power series, Dirichlet series and Euler products. We give some elucidating interpretation of some results in number theory as a manifestation of the Stieltjes integration. One of the highlights is that of Lau’s theorem on the summatory function associated to the convolution of two arithmetic functions. Toward the end we state a rather general principle–Abel-Tauber process, which consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration–differentiation process. In Section 3.8, we use Abel-Tauber process to establish interesting asymptotic expansion for the Riesz sums of arithmetic functions with best possible error estimate. The novelty of our method is that we incorporate the Selberg tyape divisor problem in this process by viewing the contour integral as part of the residual function. The novelty also lies in the uniformity of the error term in the additional parameter which varies according to the cases. Generalization of the famous Selberg Divisor problem to arithmetic progression has been made by Rieger [Rieger (1965)], Marcier [Meyer (1967)], Nakaya [Nakaya (1992)] and around the same time Nowak [Nowak (1991)] studied related subject of reciprocals of an arithmetic function and obtained asymptotic formula with the Vinogradov-Korobov error estimate with the main term as a finite sum of logarithmic terms. We shall also elucidate the situation surrounding these researches and illustrate our results by rich examples.

Our attention has been drawn to the intriguing identity for arithmetical Fourier series. On one hand, there appears the Liouville function which is a prime number-theoretic entity and on the other hand, Riemann’s example of a nowhere differentiable function. The integrated identity can be derived from the functional equation only. But to differentiate it, one needs the estimate for the error term for the Liouville function which is as deep as the PNT. [Murty (2004)] gives a very instructive proof of the quadratic reciprocity from the theta-transformation formula, which in turn as is well known (to Riemann) is equivalent to the functional equation. The situation is suggestive of Ingham’s treatment [Ingham (1964)] of the prime number theorem, i.e. first apply the A belian process (integration) and then Tauberian process (differencing) which needs more information. A big advantage of our treatment is that while the theta-function Θ(τ) is a dweller in the upper-half plane, its integrated form F(z) is a dweller in the extended upper half-plane including the real line, thus making it possible to consider the behavior under the increment of the real variable, where the integration is along the horizontal line. The elliptic theta-function θ(s) = Θ(–iτ) is a dweller in the right half-plane {σ>0}, where the integration is along the vertical line. In terms of Lambert series, there is an idea of Wintner to deal with limiting behavior of the Lambert series on the circle of convergence, i.e. radial integration. Our case corresponds to integration along an arc.

In this chapter, we shall present applications of Hilbert space theory to number theory. To enhance easy understanding of the contents, we shall provide basics of the theory in §5.1 depending mostly on [Yoshida (1967)]. In §5.2 we keep the contents in §3.4, I of Romanoff’s (and Wintner’s [Wintner (1944)]) theory in a slightly expanded form. §5.3 is the core of the chapter, presenting an accessible and partial introduction to automorphic L-functions by way of the Kuznetsov trace formula. There are a few different statements of the formula. In literature [Baker (2003)], [Miatello (1990)], [Venkov (1990)], the form stated as the Kuznetsov sum formula is the reversed form according to Iwaniec. Thus comparing these different versions in a more unified way will help the reader to comprehend the situation and at the same time it will pave a road to the coming book by Liu [Liu]. The most comprehensive one for a Fuchsian group of the first kind is given in Iwaniec [Iwaniec (1995)]. In Motohashi [Motohashi (1997)], the case of the full modular group is treated and the sum formula is the reversed form. In Iwaniec [Iwaniec (1995)], the reversed Kuznetsov formula is stated in the form of two complementary theorems, Theorem 5.27 and Theorem 5.28 below, the former the Titchmarsh series and the latter Neumann series. The full version of the Kuznetsov sum formula is Theorem 5.29 below which follows immediately from these two theorems, where (1) functional equation for the weighted Kloostermann zeta for the former and the inversion of the Hankel, Titchmarsh transforms and (2) spectral expansion of the Kloostermann zeta.

This chapter is a continuation of Chapter 6, I and we keep the material up to J-lossless factorization. If we choose the Riemann zeta-function ζ(s + 1) shifted by 1, where s = σ + jω, then we can see a very close correspondence between their region of stability and analyticity (both are σ > 0), the critical lines (σ = 0 and $\sigma =-\frac{1}{2}$, respectively), only the functional equation aspect being not clear in the former since the transfer functions are mostly just rational functions. Here, we might introduce a more sophisticated transfer function, as in the case of FOPID (Fractional Order Proportional- Integral-Differential) control, which turns out to be the Riemann-Liouville fractional integral transform (6.76), I of the input function as discussed §6.9, I. Namely, we might expect that a finer theory of transfer functions be developed which would be certain special functions whose avatars appear as rational functions (e.g. Padè approximation thereof).

The following sections are included:

• Bibliography
• Index