Theory of Numbers

publication of Gauss' paper on biquadratic residues, giving the law of biquadratic reciprocity, and his treatment of complex numbers, Jacobi found a similar law for cubic residues.

By the theory of elliptical functions, he was led to beautiful theorems on the representation of numbers by $2$, $4$, $6$, and $8$ squares. Next come the researches of Dirichlet, the expounder

of Gauss, and a contributor of rich results of his own.

Peter Gustav Lejeune Dirichlet[]88 (1805–1859) was born in Düren, attended the gymnasium in Bonn, and then the Jesuit gymnasium in Cologne. In 1822 he was attracted to Paris by the names of Laplace, Legendre, Fourier, Poisson,

Cauchy. The facilities for a mathematical education there were far better than in Germany, where Gauss was the only great figure. He read in Paris Gauss' Disquisitiones Arithmeticæ, a work which he never ceased to admire and study. Much in it was simplified by Dirichlet, and thereby placed within easier reach of mathematicians. His first memoir on the impossibility of certain indeterminate equations of the fifth degree was presented to the French Academy in 1825. He showed that Fermat's equation, $x^n+y^n=z^n$, cannot exist

when $n=5$. Some parts of the analysis are, however, Legendre's. Euler and Lagrange had proved this when $n$ is $3$

and $4$, and Lamé proved it when $n=7$. Dirichlet's acquaintance

with Fourier led him to investigate Fourier's series. He

became docent in Breslau in 1827. In 1828 he accepted a position in Berlin, and finally succeeded Gauss at Göttingen in 1855. The general principles on which depends the average number of classes of binary quadratic forms of positive and negative determinant (a subject first investigated by Gauss) were given by Dirichlet in a memoir, Ueber die Bestimmung

der mittleren Werthe in der Zahlentheorie, 1849. More recently F. Mertens of Graz has determined the asymptotic

values of several numerical functions. Dirichlet gave some

attention to prime numbers. Gauss and Legendre had given

expressions denoting approximately the asymptotic value of the number of primes inferior to a given limit, but it remained for Riemann in his memoir, Ueber die Anzahl der Primzahlen

unter einer gegebenen Grösse, 1859, to give an investigation of the asymptotic frequency of primes which is rigorous. Approaching the problem from a different direction, Patnutij Tchebycheff, formerly professor in the University of St. Petersburg

(born 1821), established, in a celebrated memoir, Sur les Nombres Premiers, 1850, the existence of limits within which the sum of the logarithms of the primes $P$, inferior to a given number $x$, must be comprised.89 This paper depends on very elementary considerations, and, in that respect, contrasts strongly with Riemann's, which