A list of books

L. E. Dickson, History of the theory of numbers (Carnegie Institution, Washington), i (1919), ii (1920), iii (1923), reprinted (Chelsea, New York, 1952).

G. H. Hardy and E. M. Wright, An introduction to the theory of num­bers (Oxford University Press, 1938, 2nd edition, 1945).

A. E. Ingham, The distribution of prime numbers (Cambridge University Press, 1932), reprinted (Stechert-Hafner, New York, 1964).

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, (2 volumes, Teubner, Leipzig, 1909), reprinted (Chelsea, New York, 1953).

J. V. Uspensky and M. A. Heaslet, Elementary number theory (McGraw­Hill, New York, 1939).

I. M. Vinogradov, An introduction to the theory of numbers (Pergamon Press, London, 1955).

Notes

Notes on Chapter I

As general references, see J. V. Uspensky and M. A. Heaslet, loco cit., Chs. 1-6; and G. H. Hardy and E. M. Wright, loco cit., Chs. 1-3.

§ 2. Theorem 2 was stated by Gauss, Disquisitiones Arithmeticae, (1801), § 16, reprinted in his Werke, i (1863), 15.

For what we call the "first proof of Theorem 2", reference may be made to E. Zermelo, Gottinger Nachrichten (new series), i (1934), 43 -44. According to Zermelo, his proof dates from 1912. See also H. Hasse, l.for Math. 159 (1928), 3-6; and F. A. Lindemann, Quarterly l. Math. (Oxford), 4 (1933), 319 - 320.

§ 3. For the "second proof of Theorem 2", see E. Heeke, Vorlesungen uber die Theorie der algebraischen Zahlen, (1923), Ch. 1. What we have called a module of integers is simply a subgroup of the additive group of integers.

For Theorem 6, see Euclid's Elements, book 7, prop. 30, given in T. L. Heath's The thirteen books of Euclid's Elements (Cambridge, 1926).

§ 5. Farey's name is associated with the Farey sequences because of Cauchy, who noticed J. Farey's statement of Theorem 7, without proof, in 1816, and published a proof himself. See A. Cauchy, Oeuvres, 2" serie, tome 6, 146. Theorems 7 and 9 seem to have been first stated and proved by C. Haros in 1802. See Dickson's History, loco cit., i, 156. The following comment by C. L. Siegel on the proof of Theorem 7 may be of interest: "Let kl-hm= 1, k>O, m>O. The homogeneous linear substitution 2=ka-hb, p.= -ma+lb of the integer variables a,b has the inverse a=21+hp., b=m2+kp.. Hence the conditions h/k~a/b~l/m, b>O, (a,b)= 1, are satisfied if and only if 2~0, p.~0, 2+p.>0, (2,p.) = 1, and then b~m+k exactly in the three cases 2,p.=0,1; 1,1; 1,0. This is independent of the notion of FR'"

§ 6. For Theorem 12, see Euclid's Elements, book 9, prop. 20. For P61ya's proof of Theorem 13, see G. P61ya and G. Szego, Auf­gaben und Lehrsiitze aus der Analysis, (1925), ii, 133, 342. The remark about allowing fo=3 is due to C. L. Siegel. The proof, by G. T. Bennett, of Euler's result that fs is divisible by 641, is given in the book by Hardy and Wright, loco cit., 15. An alternative proof is given by Kraitchik, Thiorie des nombres (Paris, 1926), ii, 221.

Notes 133

Notes on Chapter II

As general references, see Uspensky and Heaslet, loco cit., Chs. 6, 7; Hardy and Wright, loco cit., Ch. 5; and Vinogradov, loco cit., Chs. 1,2.

§ 1. The theory of congruences was developed by Gauss in his Dis­quisitiones Arithmeticae, loco cit., though Fermat and Euler were perhaps aware of some of the main results.

§ 2. For Fermat's statement of Theorem 3, in 1640, see his Oeuvres, ii, 209. Euler proved Theorem 2 in 1760. See his Opera, (l), ii, 531. See also Dickson's History, loco cit., i, Ch. 3.

§ 3. For Theorem 7, see Lagrange, Oeuvres (1868), ii, 667 -9.

Notes on Chapter III

§ 2. For the proofs of Theorems 5 and 7, see, for instance, H. Rade­macher, Lectures on elementary number theory (Blaisdell, New York, 1964), 33 - 35.

§ 3. For Theorem 6, see Lucas, TMorie des nombres (1891), i, 353 -4.

§ 4. Theorem 9 is due to A. Hurwitz, Math. Annalen, 39 (1891), 279 - 284. The proof given here is due to A. Khinchin (= A. Khintchine), Math. Annalen, 111 (1935), 631-637, and the author's attention was drawn to it by Raghavan Narasimhan. In the author's Einfiihrung in die Analytische Zahlentheorie, Springer Lecture Notes, 29 (1966), Ch. 3, a different proof was sketched, which originated with L. R. Ford, American Math. Monthly, 45 (1938), 586-601.

Notes on Chapter IV

As general references, see Uspensky and Heaslet, Hardy and Wright, and Vinogradov, loco cit.

§ 1. For the introduction of the Legendre symbol, see Legendre, Essai sur la tMorie des nombres (1798), 2nd edition (1808), § 135.

We do not consider the case p=2, since all integers are quadratic residues modulo 2.

§ 2. The first published proof (1773) of Wilson's theorem is due to Lagrange, Oeuvres, iii, 425. The theorem was first stated by Waring, Meditationes algebraicae (1770), 218, and attributed to J. Wilson. Hardy and Wright say that "there is evidence that it was known long before to Leibniz".

§ 3. Theorems 5, 6, 7 can be found in Hardy and Wright's book, loco cit., 70, 297. The proof of Theorem 7 given here is due to Hermite, Journal de Math. (1), 13 (1848), 15; Oeuvres, i, 264.

134 Notes

§ 4. Waring stated without proof that every positive integer is a sum of four squares, M editationes algebraicae (1770), 204 - 5, and Lagrange proved it the same year, see his Oeuvres, iii, 189. See also Dickson's History, loco cit., ii, Ch.8.

Notes on Chapter V

§ 1. Theorem 1, though stated by Euler, and partly proved by Legendre, was completely proved by Gauss in 1795. See P. Bachmann, Niedere Zahlentheorie (1902), i, Ch. 6, where several proofs are des­cribed.

§§ 2-3. The idea of proving Theorem 1 by means of a reciprocity formula for Gaussian sums goes back to Kronecker, Monatsber. Kgl. Preuss. Akad. Wiss. Berlin (1880), 686 - 698; 854 - 860; J. for die reine und angewandte Math. 105 (1889), 267 -268; Werke (1929), iv, 278- 300. [There is, however, a reference to a paper by Schaar (1848), on the reci­procity formula for Gaussian sums, in Lindel6fs Calcul des Residus, p. 68, as pointed out by C. L. Siegel.] It was extended to algebraic num­ber fields by E. Hecke, Gottinger Nachrichten (1919), 265-278; Werke, 235-248; and by C. L. Siegel, Gottinger Nachrichten (1960),1-16; Ges. Abhandlungen (1966), iii, 334-349. The proof given here is, in substance, Siegel's. The integral, used in the proof of Theorem 2, is of importance in the theory of the zeta-function of Riemann. See C. L. Siegel, Quellen und Studien zur Geschichte der Math. 2 (1932), 45 - 80; Gesammelte Abhandlungen (1966), i, 275.

For the evaluation of ordinary Gaussian sums by contour integra­tion, see also L. J. Mordell, Messenger of Math. 48 (1919), 54-56.

The deduction of (14) from (12) is slightly shorter here than in the author's Lecture Notes (loc. cit. Notes, Ch. 3), as a result of a comment by C. L. Siegel.

Since g( -m, -n)=g(m,n), the case m<O, n>O can be reduced to the case m>O, n<O.

Relation (21) shows that -1 is a quadratic residue of primes == 1 (mod4), and a quadratic non-residue of primes == 3 (mod4).

§ 4. Theorem 3 is due to Euler, Opera, (1), iii, 240. For the example and the remark, see Rademacher, (loc. cit., Notes,

Ch. 3), 74, 82.

Notes on Chapter VI

As general references, see Hardy and Wright, loco cit., Chs. 16-18, and Vinogradov, loco cit.

Notes 135

§ 2. The statement that r(n) = O(nE), for every £ > 0, is equivalent to the statement that r(n)=o(nE), for every £>0.

For Theorem 1, see Gauss, Werke, (ii), 272 - 5.

§ 3. For the proof of Theorem 4, see P6lya and Szego (loc. cit., Notes, Ch. 1), ii, 160-1, 386.

For Theorems 5 and 6, see Hardy and Wright, loco cit., 259. Theorem 9 was proved by Dirichlet in 1849, see his Werke, ii, 49 - 66. G. Voronoi's improvement of the error-term is given in Ann. Sci.

Ecole Norm. Sup. (3), 21 (1904), 207 - 267; 459 - 533. That the error­term is not O(N1/4) was proved by Hardy, Proc. London Math. Soc. (2) 15 (1916),192-213.

§ 4. For the history of Mersenne numbers, and of perfect numbers, see Dickson, loco cit., i, Chs. 1 - 2.

§ 5. For Theorems 15 and 19, see A. F. Mobius, J. for die reine und angewandte Math. 9 (1832), 105 -123; Werke (1887), iv, 589 - 612. See also Landau's Handbuch, loco cit., §§ 150-152. Theorems 16 and 17 were proved by Dedekind, J.for die reine und angewandte Math. 54 (1857), 21, and by Liouville, J. de Math. pures et appliquees, (2) 2 (1857), 111, at about the same time.

§ 6. For Theorem 20, see Landau's Handbuch, loco cit., § 59. Theo­rem 22 is due to F. Mertens, J. fur die reine und angewandte Math. 77 (1874),290-291, and is given in Landau's book, § 152.

00

The evaluation of L f.1(n) n - 2, without the use of Euler's identity n=l

(proved later in Chapter VII, § 4) is a result of a comment by Raghavan 00

Narasimhan. For a proof of the formula L n - 2 = 1[2/6, see, for in-"=1

stance, K. Knopp, Theory and application of infinite series (1951), 237, 323,376.

Notes on Chapter VII

As general references, see Landau's H andbuch, loco cit., §§ 12 - 28, and Ingham's book, loco cit., Ch. 1.

§ 1. For Theorem 1, see Euler, Opera (1),8, § 279; (1), 14,216-244.

§ 2. Theorem 3 is due to Chebyshev, Oeuvres, i, 49 - 70.

§ 3. S. S. Pill ai's proof of Theorem 4 is given in Bull. Calcutta Math. Soc. 36 (1944), 97 -99; 37 (1944), 27. See also Landau's Handbuch, loco cit., § 22.

§ 4. Theorem 7 is due to Chebyshev, Oeuvres, i, 27 -48. See Ingham's book, loco cit., 16 - 21. Euler used the formal identity.

136 Notes

§ 5. Theorem 8 is due to F. Mertens, J. for die reine und angewandte Math. 78 (1874), 46 - 62. See Ingham's book, loco cit., 22.

Stirling's formula is given, for instance, in the book by E. C. Titch­marsh, The theory of functions (Oxford, 1932), 2nd edition (1939), § 1.87.

Notes on Chapter VIII

§§ 1-4. Weyl's theorems were proved by him in Math. Annalen, 77 (1916), 313 - 352. An exposition using the notion of "discrepancy" is given by 1. W. Cassels, An introduction to Diophantine approximation (Cambridge, 1957), Ch. 4.

§ 5. Kronecker proved his theorem in the Berliner Sitzungsberichte (1884); see his Werke, iii (i) 47 -110. For further developments, see J. F. Koksma, Diophantische Approximationen, Ergebnisse der Math. Band iv, Heft 4 (1937).

H. Bohr's proof of Theorem 8 is given in J. London Math. Soc. 9 (1934),5-6. See also Hardy and Wright, Ch. 23.

Notes on Chapter IX

As general references, see Minkowski's Geometrie der Zahlen, lSI edi­tion (1896), and his Diophantische Approximationen (1927). See also the Lecture Notes on the Geometry of Numbers by C. L. Siegel (New York University, 1945).

§ 2. Theorem 1 is true without the assumption that the set S is bounded. For if it is unbounded, with measure V(S) > 2n , one can take its intersection with a cube KM given by IXkl <M, 1 ~k~n, and if M is sufficiently large, then S M = S n K M will be a bounded set satisfying the required conditions, because of the countable additivity of Lebesgue measure.

We do not seek to give the optimal hypotheses here since we do not wish to go into questions of measurability in greater detail. The formulation and proof of Theorem 3 support this line.

Minkowski proved Theorem 3 in 1891, see his Gesammelte Abhand­lung en, i, 264.

Siegel's proof of his formula (8) is given in Acta Math. 65 (1935), 307 -323.

The lemma which appears between Theorems 2 and 3 is due to G. D. Birkhoff, as stated by Blichfeldt, Trans. American Math. Soc. 15 (1914), 230. See also an Appendix in Cassels's book (loc. cit., Notes, Ch. 8).

In Theorem 2 we use the fact that a closed set in W is Lebesgue measurable.

Notes 137

Minkowski (loc. cit.) shows that a bounded convex set in R n has a volume in the sense of Jordan. See his Geometrie der Zahlen, 50-60; also his Theorie der konvexen Korper, Ges. Abh. 2, 142 -143; Blaschke's Kreis und Kugel, (Leipzig, 1916), 57.

If a convex set S has Lebesgue measure V(S), 0< V(S)< 00, then it is bounded. See J. W. S. Cassels, An introduction to the geometry of numbers (Springer 1919), 109.

Notes on Chapter X

As a general reference, see Landau's Handbuch, loco cit., §§ 95 -103. See also C. L. Siegel, Lectures on Analytic Number Theory (New York University, 1945).

§ 5. The main theorem of this chapter, namely Theorem 8, was first proved by Dirichlet in 1837, see his Werke (i), 307 - 342. An elementary proof was given by Mertens, Wiener Sitzungsberichte, 106 (1897), 254-286. A proof of the theorem by a new elementary method is due to A. Selberg, Annals of Math. (2) 50 (1949), 297 - 304; Canadian J. of Math. 2 (1950),66-78. Another elementary proof is due to H. Zassen­haus, Comm. Math. Helvetici, 22 (1949),232-259.

Notes on Chapter XI

As a general reference, see Landau's Handbuch (loc. cit.) including the Appendix by P. T. Bateman, pp. 929 - 931, which gives a history of the proof of the prime number theorem as an asymptotic relation. The idea of connecting the behaviour of n(x) with the properties of ((s), where s is a complex variable, and, in particular, with the location of its non-real zeros, originated with Riemann, Ober die Anzahl der Prim­zahlen unter einer gegebenen GrojJe, Monatsberichte der Preuss. Akad. der Wissenschaften (Berlin, 1859 -1860),671- 680; Werke (1 sl edition, 1876), 136-144; (2nd edition, 1892), 145-155.

§ 1. The first proof of the prime number theorem was given by J. Hadamard, Bull. de la Soc. Math. de France, 24 (1896),199-220; and by c.-J. de la Vallee Poussin, Annales de la Soc. sci. de Bruxelles, 202

(1896), 183 - 256. For a clear presentation of the classical proof, see Ingham's book (loc. cit.), Ch. 2.

§ 2. For the theorem of Wiener-Ikehara, see S. Ikehara, J. Math. Phys. Mass. Inst. Tech. 10 (1931), 1-12; N. Wiener, Annals of Math. 33 (1932), 1-100, 787; and N. Wiener, The Fourier integral (Cambridge, 1933), § 19. The theorem is true with weaker hypotheses, but for the deduction of the prime number theorem, it does not make much dif­ference what form we consider.

138 Notes

The proof of the Wiener-Ikehara theorem given here, which does not use Wiener's general Tauberian theorem, is substantially that of S. Bochner, Math. Zeit. 37 (1933), 1-9, as simplified by E. Landau, Ber­liner Sitzungsberichte (1932),514-521, and by Bochner in his Lectures on Fourier Analysis (Princeton University, 1936). It is the same as the proof given in the author's Lectures on the Riemann zeta-Junction (Tata Institute of Fundamental Research, Bombay, 1953). A proof of the prime number theorem by a new elementary method has been given by A. Selberg, Annals oj Math. (2) 50 (1949), 305 - 313.

Subject index

Abel's summation formula 78 arithmetical function 14;

completely multiplicative - 76; multiplicative - 14; - r(n) 45; - R(N) 46; - d(n) 47; - D(N) 50; - (J(n) 54; - J1(n) 55; - A(n) 57; - cp(n) 13; - q,(N) 59; - n(n) 63; - 9(n) 64; - t/!(n) 64;

Bertrand's postulate 71 Birkholrs lemma 100 Character: - of an abelian group -

107; - modulo m 110; principal- 107, 110

Chebyshev's: - functions 64; - inequality 74; -lemma 68; - theorem 67

composite number 1 congruences 11 ; sum, difference,

product of - 11 Congruent: - modulo m 11;

-modulo 1 84 Convergence: abscissa of - 113;

abscissa of absolute - 114; half-plane of - 113; line of -113; strip of conditional- 114

Dirichlet's: - formula for D(N) 53; - theorems 84, 120; - L-functions 117

Dirichlet series 78, 111; coefficients of - 78, 111; formal product of - 116; product of - 116; uniqueness of - 116

discrepancy 85; - modulo 1 87 divisibility 1 divisor 1; greatest common - 3;

- function 47

Euclid's theorem 4 Euler's: - constant 51; - criterion

27; - function cp 13; - identity 76; - theorem 13

Farey: - fraction 6; - sequence 6 Fermat number 10

Fermat's theorem 13 fraction 6; irreducible - 6;

proper- 6; reduced- 6 fractional part 84

Gaussian sum 34; generalized-34

group: abelian - 107; cyclic-107; generator of - 107

Hadamard's theorem 123 Hurwitz's theorem 23; Khinchin's

proof of- 23

integral part 18 interval function 85

Kronecker's theorem 91; Bohr's proof of - 93

Lagrange's theorem: - on congruen­ces 16; sums of squares 31

Landau's theorem 115 lattice 101; determinant of a-

101 lattice point 45; - function r(n)

45 Legendre symbol 26 linearly independent 91

von Mangoldt's function 57 mediant 8 Mersenne: - number 55;

-prime 55 Mertens's: - formulae 81;

- theorem 59 Minkowski's theorem 98; Siegel's

proof of - 98 Mobius's function 55 Mobius inversion formula: first - 56;

second - 58 module 3; the trivial- 4 multiple 1; integral - 1; least

common- 5

Orthogonality relations 110

Polya's theorem 10 perfect number 54

140 Subject index

prime: - number 1; - residue class 12; relatively - 3

prime number theorem 122 principal character 107, 110

quadratic: - residue 26; - non-residue 26; - reciprocity law 34

quadratic form: positive definite-104; determinant of a-I 04

quotient 1

remainder 1 representation: imprimitive - 30;

primitive 30 residue class 11 ; prime - 12 residue system: complete - 12;

complete prime - 12

Riemann's zeta-function 107

set: convex - 97; symmetric -97; translate of a - 97

Siegel's formula 99 standard form 2 Stirling's formula 81 summatory function 45

uniformly distributed 85; -modulo 1 86

unique factorization theorem 2

de la Vallee Poussin's theorem 123

Weyl's theorems 87 Wiener-Ikehara theorem 124 Wilson's theorem 27

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