Section 1.2 Special Values of Zeta Functions

Several generalizations of the Riemann Zeta function have been introduced and studied so far, which will be discussed in more detail in the next section. These functions are often referred to as \(L\)–functions or zeta functions. Among many other applications to number theory, special values of \(L\)–functions are one of the most important themes to study because they encode lots of arithmetic information. The foundational result in this regard is the Analytic Class Number Formula (see [29, Chapter 7]), first proved by Dedekind [11]. The simplest example: the special value \(\zeta(0)=-1/2\) encodes the arithmetic information that \(\Z\) is a Unique Factorization Domain or UFD for short. As mentioned before, Euler used the fact that \(\zeta(1)\) diverges to show [13, Theorem 7] that there are infinitely many primes. A slightly more advanced application: the special value \(\zeta(-1)=-1/12\) implies that the Milnor \(K\)-group \(K_2(\Z)\) of the integers has cardinality \(2\text{.}\) This is a special case of the Birch-Tate conjecture, which will be discussed later. In particular, we know since Euler that \(\zeta(-n)\) is a rational number for odd positive integers \(n\text{.}\)

In this thesis, we are concerned about certain congruences of special values of zeta functions. Our method of proof is to use a \(p\)-adic analogue of Dirichlet \(L\)–functions. The very existence of \(p\)-adic \(L\)–functions was heralded in a congruence of special zeta values due to Ernst Kummer:

We owe Kummer the discovery of the striking connection between the arithmetic of cyclotomic fields and the special values of the Riemann zeta function. But that was just the tip of an enormous iceberg! Over the years, after Kenkichi Iwasawa’s groundbreaking works, these connections have culminated in a deep area of research known as Iwasawa Theory. The Iwasawa Main Conjecture, now a theorem due to Mazur and Wiles [30], is the deepest result we know about the arithmetic of cyclotomic fields.