1911 Encyclopædia Britannica/Number/Number of classes

39. Number of classes. Class-number Relations.—It appears from Gauss's posthumous papers that he solved the very difficult problem of finding a formula for $h(\mathrm {D} )$, the number of properly primitive classes for the determinant $\mathrm {D}$. The first published solution, however, was that of P. G. L. Dirichlet; it depends on the consideration of series of the form $\Sigma (ax^{2}+bxy+cy^{2})^{1-s}$ where $s$ is a positive quantity, ultimately made very small. L. Kronecker has shown the connexion of Dirichlet's results with the theory of elliptic functions, and obtained more comprehensive formulae by taking $(a,b,c)$ as the standard type of a quadratic form, whereas Gauss, Dirichlet, and most of their successors, took $(a,2b,c)$ as the standard, calling $(b^{2}-ac)$ its determinant. As a sample of the kind of formulae that are obtained, let $p$ be a prime of the form $4n+3$; then

$h(-4p)=\Sigma \alpha -\Sigma \beta ,\quad h(4p)\log(t+u\surd p)=\log \Pi \left(\tan {\frac {b\pi }{4p}}\right)$

where in the first formula $\Sigma \alpha$ means the sum of all quadratic residues of $p$ contained in the series $1,2,3,\dots {\tfrac {1}{2}}(p-1)$ and $\Sigma \beta$ is the sum of the remaining non-residues; while in the second formula $(t,u)$ is the least positive solution of $t^{2}-pu^{2}=1$, and the product extends to all values of $b$ in the set $1,3,5,\dots (4p-1)$ of which $p$ is a non-residue. The remarkable fact will be noticed that the second formula gives a solution of the Pellian equation in a trigonometrical form.

Kronecker was the first to discover, in connexion with the complex multiplication of elliptic functions, the simplest instances of a very curious group of arithmetical formulae involving sums of class-numbers and other arithmetical functions; the theory of these relations has been greatly extended by A. Hurwitz. The simplest of all these theorems may be stated as follows. Let $\mathrm {H} (\Delta )$ represent the number of classes for the determinant $-\Delta$, with the convention that ${\tfrac {1}{2}}$ and not $1$ is to be reckoned for each class containing a reduced form of the type $(a,0,a)$ and ${\tfrac {1}{3}}$ for each class containing a reduced form $(a,a,a)$; then if $n$ is any positive integer,

${\underset {\kappa =0,\pm 1,\dots }{\Sigma }}\mathrm {H} (4n-\kappa ^{2})=\Phi (n)+\Psi (n)\qquad (-2\surd n\leq \kappa \leq 2\surd n)$

where $\Phi (n)$ means the sum of the divisors of $n$, and $\Psi (n)$ means the excess of the sum of those divisors of $n$ which are greater than $\surd n$ over the sum of those divisors which are less than $\surd n$. The formula is obtained by calculating in two different ways the number of reduced values of $z$ which satisfy the modular equation $\mathrm {J} (nz)=\mathrm {J} (z)$, where $\mathrm {J} (z)$ is the absolute invariant which, for the elliptic function ${\mathfrak {p}}(u;\ g_{2},g_{3})$ is ${g_{2}}^{3}\div ({g_{2}}^{3}-27{g_{3}}^{2})$, and $z$ is the ratio of any two primitive periods taken so that the real part of $iz$ is negative (see below, § 68). It should be added that there is a series of scattered papers by J. Liouville, which implicitly contain Kronecker's class-number relations, obtained by a purely arithmetical process without any use of transcendents.