1911 Encyclopædia Britannica/Number/Number of classes

39. ''Number of classes. Class-number Relations.''&mdash;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,
 * $$\underset{\kappa=0,\pm 1, \dots}{\Sigma} \mathrm{H}(4n-\kappa^2) = \Phi(n)+\Psi(n) \qquad (-2\surd n \le \kappa \le 2\surd n)$$

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, &sect; 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.