## Hardy and Wright, Chapter 8 (Second Part)

Picking up where we left off last week, we finished chapter 8 today. Most of the time was spent trying to trace through the proofs of the various statements, so I won’t go into too much detail here about that. Many of the proofs had the same flavor, cleverly grouping terms in a polynomial, or setting corresponding coefficients equal in two different representations of a polynomial.

When I had first read the chapter, I didn’t pay too close attention to some of the later sections, for example the section on Leudesdorf’s Theorem (generalizing Wolstenholme’s), and the “Further consequences of Bauer’s Theorem“. However, during our meeting we worked through most of Leudesdorf’s theorem, and we were able to gain some appreciation for the various cases (specifically, why they arise).

One of the theorems in the sections we kinda glossed over was the following (Theorem 131 in the book): If $p$ is prime, and $2v, then the numerator of $S_{2v+1}=1+\frac{1}{2^{2v+1}}+\cdots+\frac{1}{(p-1)^{2v+1}}$ is divisible by $p^2$. I noted that this $S_{2v+1}$ is a partial sum for $\zeta(2v+1)=\sum_{n=1}^{\infty} n^{-(2v+1)}$ (Wikipedia, Mathworld). Eric wondered if perhaps they were thinking about this sum as a generalization of this $\zeta$ function to some finite field, but the modulus of $p^2$ didn’t fit that entirely. Eric also reminded us that closed forms for $\zeta(2v)$ can be found, while closed forms for $\zeta(2v+1)$ are not known.