DEFINITION 6: A finite number is a number which possesses any property which belongs to o, and also to the successor of any number which has the property.
We can now prove all five of Peano's axioms, i.e. all the ordi- nary arithmetic of the finite integers, and so all the arithmetic of real and complex numbers, without assuming anything beyond the laws of logic and the axiom of infinity. Both geometry and arithmetic are purely formal, or logical, in their method of deduction. But the ordinary meaning of geometry makes it a branch of physics, while the ordinary meaning of arithmetic makes it the development of logic itself.
For the detailed development of arithmetic on a logical basis see G. Peano, Formulaire de mathematiques (1908) ; G. Frege, Die Grundlagen der A rithmetik (1884) and Grundgesetze der Arithmetik (1893, 1903); B. Russell, The Principles of Mathematics (1903) and Introduction to Mathematical Philosophy (1919-); A. N. Whitehead and B. Russell, Principia Mathematica (1910, 1911, 1913).
(J- N.) (2.) THEORY or NUMBERS
In the article " Number " (19.847) an excellent summary is given of the classical theory. Modern mathematics has seen the rise of a new theory, the " analytic " theory, which has developed with astonishing rapidity, and has almost monopo- lized the attention of arithmeticians.
(a) Theory of Primes. The modern developments of the theory of numbers depend in the main on the application to the theory of the ideas of the theory of functions of a complex variable (see 11.301). It was in the theory of the distribution of primes that these ideas first bore fruit.
It is usual to write ir(x) for the number of primes less than x. It has been known since Euclid that the number of primes is infinite, that is to say that TT(X) tends to infinity with x. The central problem of the theory has been the determination of the order of magnitude of ir(x) when x is large, and its solution is embodied in the Prim- zahlsatz, or " prime number theorem," expressed by the formula
-r
log*
where the symbol ' > expresses the fact that the ratio of the two functions tends to unity. This theorem, conjectured by A. M. Legendre (1798) and C. F. Gauss (about 1792), was first proved by J. Hadamard and Ch. J. de la Vallee Poussin in 1896. The real founder of the modern theory, however, was B. Riemann, who, in a famous memoir published in 1859, first indicated the road along which subsequent research has progressed. Riemann did not prove the prime number theorem; strangely enough, he did not mention it, his object being to obtain, not an asymptotic formula for v(x) but an exact expression in the form of an infinite series. Nor did Riemann attain the goal at which he aimed, his analysis, profound and beautiful as it is, being altogether incomplete and inconclusive. But it was Riemann who first recognized where the key to the solu- tion lay, viz. in the study of the " Riemann zeta-f unction "
(where n = i, 2, 3, . . . and p runs through the series of primes), considered as a function of the complex variable 5. Riemann estab- lished some, and conjectured others, of the properties of f(s); one famous conjecture, that all the complex zeros of $(s) lie on the line a = J, remains unsettled to this day.
Riemann's memoir bore no fruit for over 30 years, when the way was cleared by the researches of Hadamard in the theory of analytic functions (see FUNCTION, 1 1 .301 seq.). These researches led Hadamard himself, de la Vallee Poussin, and other writers, to a proof not only of the prime number theorem but of very much more. Thus de la
Vallee Poussin proved that the logarithm integral Li*= I : -
represents ir(x) with an error of lower order than x(log *)"*, where k is any number however large. He also investigated the distribu- tion of primes of a linear form am+b or a quadratic form am>+bm +c, where a, b, c are integers without common factor, showing, for example, that the primes are, on the average, equally distributed between the various arithmetical progressions am + i, am+2, ... ., as had been conjectured long before by P. G. Lejeune Dirichlet. There is a corresponding theory for the " prime ideals " of the " corpus " associated with any algebraic number. The ana- logue of Riemann's zeta-function was discovered by R. Dedekind, but it is only recently that, in the hands of E. Hecke and E. Landau, the development of the theory has been pushed to a point corre- sponding with that of the ordinary theory.
The outstanding unsolved problem of the theory is that of the determination of the order of the difference ir(x) Lix. This problem is bound up essentially with Riemann's unproved hypothesis con- cerning the zeros of f(s). If Riemann's hypothesis is true, the max-
imum order of the difference differs from that of V* by logarithmic factors only. In any case the difference assumes values of either sign which tend to infinity with x. This theorem, proved by J. E. Littlewood in 1914, disposes of the old conjecture of Gauss and B. Goldschmidt that ir(x) is always less than Li(x).
Apart from applications to the theory of primes, there is a large literature connected with the pure theory of f(s). It has been shown by H. Bohr, E. Landau and F. Carlson that (to put it roughly) nearly all the zeros lie very near the critical line; by G. H. Hardy and J. E. Littlewood that (equally roughly) a considerable propor- tion lie actually on it. But the hypothesis itself remains unproved.
(b) Additive Theory. The "additive" theory of numbers in- cludes Combinatory Analysis (see 6.752), Partitions (see 19.865), the theory of the representation of numbers by sums of squares, cubes, or higher powers, and so forth.
The central problem is that of determining (exactly or approx- imately) the number of representations of an arbitrary positive integer n in the form 01+02+ . . . +o a , where the o's are num- bers of some special type (e.g. squares), and i may be fixed or un- restricted, according to the particular problem envisaged. There is a fundamental difference between the ' additive " theory and what may be called the " multiplicative " theory, in which the central idea is that of the resolution of a number into prime factors. Analyti- cally, this difference expresses itself as follows: the multiplicative theory depends on the theory of " Dirichlet's series " of the type 2on', the additive theory on that of power series Son*". A great deal of the additive theory is purely algebraic, and is intimately bound up with the theory of elliptic functions. This side of the theory (founded by L. Euler) has been developed to a high pitch by English mathematicians, notably A. Cayley, J. J. Sylvester, and P. A. Mac- Mahon, while more recently the methods of complex function theory have been applied to the theory and an " analytic additive " theory has been founded. Among many curious results we may mention the theorem of S. Ramanujan, that the numbers of the unrestricted partitions of numbers of the forms 5771+4, 7i+5 and nm+6 are divisible by 5, 7 and II respectively.
One of the most remarkable problems of the additive theory is " Waring's Problem." It was asserted by E. Waring (1782) that any number re is the sum of at most 4 squares, 9 positive cubes, 19 fourth powers, and, generally, g(k) powers, where g(k) is a number de- pending on k alone and not on n. This problem (in so far as it simply asserts the existence of some such number g(k}, was solved by D. Hilbert in 1909. J. L. Lagrange (1774) proved that g(2)=4 (any number is the sum of 4 squares, and some numbers not of less), and E. Wieferich (1909) that g(3)=9 and g(4) ^ 37. Only a finite number of numbers (probably only 23 and 239) require more cubes than 8 (E. Landau, 1908), while an infinite number require 4 at least ; and only a finite number of numbers require more than 21 fourth powers (G. H. Hardy and J. E. Littlewood, 1921), while an infinite number require 1 6 at least ; and asymptotic formulae for the number of representations have been found; but our knowledge of this field is still extremely incomplete.
The " empirical theorem " of Chr. Goldbach, that every even number is the sum of two primes, has also received a considerable amount of attention, but is still unproved. Among other unsolved problems of the same character may be mentioned that of proving the existence of an infinity of primes of the form n 2 + l or (more generally) om 2 +&w+c. This problem is not to be confused with the problem of primes am 2 -\-bmn -\-cri*, solved by de la Vallee Poussin.
(c) Miscellaneous Investigations. The work of Dirichlet and L. Kronecher on the approximation of irrational numbers by rationals has led to extensive investigations lying on the border line between arithmetic and analysis, developed above all by H. Minkowski under the titles of Diophantische Approximation and Geometrie der Zahlen. The central idea in this theory is that of the lattice (Ciller).
A lattice point (Gitterpunkt) in space of any number of dimensions is a point with integral coordinates, and most difficult and fascinating problems arise when we consider the number of lattice points which lie within a volume of specified form in w-dimensional space. Thus Minkowski proved that any convex figure in space of two dimensions with symmetry about a centre, its centre at a lattice point, and of area 4, includes other lattice points besides its centre; with a whole series of corresponding theorems concerning more general configura- tions. Another class of lattice-point problems is exemplified by the " circle " problem of Gauss and W. Sierpinski, that of determining approximately the number of lattice points inside the circle x 2 +y 2 = n when n is large. A first approximation is naturally given by irn, the area of the circle, but the estimation of the error is a problem of exceptional difficulty. This problem and the analogous problem for the hyperbola xy = n (Dirichlet's divisor problem) were connected with the theory of f(s) [see (a) supra] by Landau. These problems also are susceptible of manifold generalization. And in all these problems, we observe the dominating and irresistible tendency of modern higher arithmetic, the tendency to abandon its ancient tradi- tion of isolation and assimilate itself so far as possible to the theory
