of functions, in order to utilize the immensely powerful weapons which the latter theory alone can provide.
There is one famous problem in which no such reduction of arithmetic to analysis has been effected. " Fermat's last theorem asserts that there is no integral solution of x n -\-y n =z n (other than the trivial solution x = z, y = o) for any value of n greater than 2. It was the attempt to prove this theorem that led to the whole development of the theory of algebraic numbers; but, in spite of the widespread attention which it has excited, and the extreme impor- tanceof the general theories of which it has been the starting point, the theorem itself remains unproved, though important additions have been made recently to our knowledge by A. Wieferich, D. Miri- manov, L. E. Dickson, and H. S. Vandiver. Thus Wieferich proved that the theorem holds for odd prime values of n, and values of x, y, z, not divisible by n, unless 2" l I is a multiple of n'.
One old conjecture has been definitely disposed of. Mersenne asserted that 2" I, where n is a prime not exceeding 257, is prime when, and only when, n = I, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257. This statement contains at least four errors, relating to the values 61, 67, 89, 107; and it need no longer be taken seriously.
AUTHORITIES. An indispensable work for the serious student of higher arithmetic (on any of its sides) is L. E. Dickson, History of the Theory of Numbers, 1920-1. This work is not, however, specially concerned with the analytic theory.
For general accounts of the theory of primes see Encycl. des Sc. Math. i. 17 (" Propositions transcendantes de la theorie des nom- bres," by J. Hadamard and E. Maillet: the article by P. Bachmann in the first German edition is inadequate, but the third edition, in preparation in 1921, was to include an account of the theory, as it stands to-day, by H. Cramer) ; E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen (1909), and Einfiihrung in die elemen- tare und analytische Theorie der algebraischen Zahlen (1918)..
For the additive theory see P. A. MacMahon, Combinatory Analysis (1915-6), and An Introduction to Combinatory Analysis (1921); P. Bachmann, Niedere Zahlenlheorie, II. (Additive Zah- lentheorie) (1910) ; G. H. Hardy, Some Famous Problems of the Theory of Numbers (1920).
For Fermat's last problem see P. Bachmann, Das Fermatproblem (1917); L. J. Mordell, Four Lectures on Fermat' s Last Problem (1921). Comparatively little of recent work is accessible in a connected form, and the study of the original memoirs is indispensable.
(G. H. H.) (3.) THEORY or SERIES
The most striking modern developments in the theory of series (see 24.668; 10.753; I2 -956) have also been suggested by the development of the theory of functions.
The theory of functions of a real variable has been revolu- tionized by the ideas of E. Borel and H. Lebesgue, and this revolution has inspired a corresponding revolution in the theory of Fourier's series and " series of orthogonal functions " generally.
A system of 'functions m (x) (m = i , 2, 3, . . . ) is said to be orthogonal \i \ a m (x)4> n (x}dx = o(mri) . . . (i).
The simplest examples are obtained by taking m (x) to be cos mx or sin mx and the interval (a, i) to be (o, 211-) ; or m (x) to be Legen- dre's polynomial P m (x) and (a, 6) to be ( 1, i). There is then a simple procedure by which we may endeavour to expand an ar- bitrary function f (x) in the form of a series 2a m < m (x), viz. by mul- tiplying this series by $, (x) and integrating over the interval (a, 6) : the formula thus suggested is
A more accurate analysis of this procedure raises a multitude of profoundly interesting and difficult questions. On the one hand we may start from a series with arbitrary coefficients a m , and inquire whether there exists a function which stands to it in the relation expressed by the equation (2). In particular, given a trigonometrical series 2a m cos mx or 26 m sin mx or, more generally, 2(a m cos mx+b m sin mx), with arbitrary coefficients, we may ask whether it is a Fourier's series, that is to say, whether there is a function f(x) such that a m and b m are given by Fourier's integral formulae. On the other hand, we may start not from an arbitrary series but from an arbitrary function f(x), form the coefficients (a m or 6 m ) by Fourier's formulae or the more general formulae (2), and then inquire whether the formal development thus obtained is convergent, and whether, if convergent, it represents the function/(x) and so forth.
The problems thus raised are among the most difficult of modern mathematics; and a very cursory examination of them is enough to show that the methods of the older analysis are not sufficiently powerful for their solution. It is essential that we should enlarge our conceptions, on the one hand, by taking account of the modern gener- alizations of the notion of an integral, and, on the other, by adopting a broader view as to what is meant by the " sum " of an infinite series.
The modern theory of functions of a complex variable (see 1 1 . 301 ) points to the same conclusion. A function f(z) of the complex variable z, regular forz = zo, is defined throughout a certain circle whose centre is ZD by a power-series 2o (z zo) n ; but the region of
existence of the function is very generally more extensive than the circle of convergence of the series; and this fact has led, during the last generation, to a mass of work on the problem of "analytic continuation." This problem is that of discovering analytic repre- sentations of the function, whether by integrals, or by continued fractions, or by series of a different form, which are valid throughout a wider region than that in which it is represented by the original power series. Here also we are confronted by the need for a scientific theory of divergent series.
There are passages in the older analysts (e.g. in L. Euler), which suggest a half-conscious anticipation of modern ideas. But it is roughly true to say that they did not concern themselves with the precise meaning of the infinite series of which they made such effec- tive use. A. L. Cauchy and N. H. Abel were the first to give a precise definition of the "sum" of a series ao+ai+Oj-f-. . . or 2o, viz. as the limit of i n = ao+ai+ +a when n tends to infinity (>). Such a series as I I, I . . . has then no sum, for s n is alternately I and o; and it was the tendency, for many years after Cauchy and Abel, to banish such series from analysis entirely. A school of mathematicians survived, among whom one may cite A. de Morgan, who viewed this tendency with obvious discontent, but there was no escape from the conclusion that the followers of Cauchy and Abel were right. It is impossible to say " the sum of So is so-and-so " except after framing an accurate definition of " sum " ; the definition of Cauchy and Abel was the only definition; and, until some new and wider definition was offered, that was the end of the matter:
We may define the meaning of a mathematical word or symbol as we please, provided only that the definition is free from contradiction. Given a sequence of numbers ai, 02, . . . we may associate with the sequence a number J in any manner that we please, and we may say, if we like, that s is the " sum " of the series. We might say, for in- stance, that the " sum " of every infinite series is, by definition, zero. This definition would be perfectly legitimate; but futile, because it would reduce all equations involving infinite series to the trivial form = 0; and confusing, because it would conflict with Cauchy 's definition. Cauchy's definition is only one among many, but it is admittedly the most important, and a new definition is only likely to be of value if it is consistent with the standard definition. It must satisfy what is called the condition of consistency; it must apply to all convergent series, and give a " sum " equal to their sum in the ordinary sense. Its value for analysis will then be measured by the extent and importance of the class of non-convergent series to which it attributes a " sum."
The simplest and most important of the definitions which have been given is that of the " first arithmetic mean." Suppose that Sn = ao+ai + _. . . +0n and = (st>+Si + . . . -Hn)(n+i), the arithmetic mean of the first n-\-l values of s n . If s n tends to a limit, a n tends to a limit also, and the two limits are the same; but a n may tend to a limit when s n does not. For example, if a n = ( 1), s 2n i and s 271+ i = o, and s n does not tend to a limit; but n tends to the limit j. If now we agree to call the limit of n , whenever it exists, the " sum " of the series 2a n , our new definition is in perfect accord with Cauchy's definition, but is applicable to an extensive class of series for which Cauchy's definition fails. It therefore fulfils the conditions required for a theory of divergent series.
The most striking illustration of the importance of these ideas is to be found in the theory of Fourier's series (see 10.753). The Fourier's series of a continuous function f(x) is not necessarily con- vergent ; further conditions on/(x), of a much more artificial charac- ter, are required to insure convergence. It was, however, shown by L. Feier that the Fourier series of any continuous function is " summable " by the procedure indicated above; that is to say, that the arithmetic mean a n tends to a limit equal to the value of the function ; and this fundamental result has been the starting point of a mass of modern research.
Another important definition attributes to the series as " sum " the value of the limit of the power series 2a n x" when x tends to I through positive values less than I. A third (of particular importance in complex function theory) was advanced by Borel; and all of these definitions have given birth to a multitude of still more general definitions.
AUTHORITIES. For the general theory of divergent series see E. Borel, Lemons sur les series divergentes (1901) ; T. J. I' A. Bromwich, Introduction to the Theory of Infinite Series, ch. x. (1908) ;G. H. Hardy and M. Riesz, The General Theory of Dirichlet's Series (1915). For the theory of Fourier's series, H. Lebesgue, Lemons sur les series trigonometriques (1912); Ch. J. de la Vallee-Poussin, Cours d'analyse infinitesimals, 2nd ed., vol. ii. (1916); E. H. Hobson, The Theory^ of Functions of a Real Variable (1907, 2nd ed. in course of publication in 1921). The general theory of series of orthogonal functions is, for the most part, still only to be read in the original memoirs, or in works on the theory of integral equations. (G. H. H.)
(4.) THEORY or FUNCTIONS
The theory of functions (see 11.301, 14.53) h & S two great branches, the real and the complex theories. Recent advances in the complex theory, important as they are, have been of too
