Introduction to analytic number theory apostol

Undergraduate Texts in Mathematics Edilors F W Gehring P R Halmos Advisory Board C DePrima I Herstein J Kiefer W LeVeque Tom M Apostol Introduction to Analytic Number Theory Springer-Verlag New York 1976 Heidelberg Berlin Tom M Apostol Professor of Mathematics California Institute of Technology Pasadena California 91 I25 AMS Subject Classification 10-01, 1OAXX Library of Congress (1976) Cataloging in Publication Data Apostol, Tom M Introduction to analytic number theory (Undergraduate texts in mathematics) ” Evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years.” Bibliography: p 329 Includes index Numbers, Theory of Arithmetic functions Numbers, Prime I Title 512’.73 75-37697 QA24l A6 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag @ 1976 by Springer-Verlag New York Inc Printed in the United States of America ISBN o-387-90163-9 Springer-Verlag New York ISBN 3-540-90163-9 Springer-Verlag Berlin Heidelberg iv Preface This is the first volume of a two-volume textbook’ which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years It provides an introduction to analytic number theory suitable for undergraduates with some background in advanced calculus, but with no previous knowledge of number theory Actually, a great deal of the book requires no calculus at all and could profitably be studied by sophisticated high school students Number theory is such a vast and rich field that a one-year course cannot justice to all its parts The choice of topics included here is intended to provide some variety and some depth Problems which have fascinated generations of professional and amateur mathematicians are discussed together with some of the techniques for sc!ving them One of the goals of this course has been to nurture the intrinsic interest that many young mathematics students seem to have in number theory and to open some doors for them to the current periodical literature It has been gratifying to note that many of the students who have taken this course during the past 25 years have become professional mathematicians, and some have made notable contributions of their own to number theory To all of them this book is dedicated ’ The second volume is scheduled to appear in the Springer-Verlag Series Graduate Texts in Mathematics under the title Modular Functions and Dirichlet Series in Number Theory V Contents Historical Chapter Introduction The Fundamental 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 of Arithmetic Introduction 13 Divisibility 14 Greatest common divisor 14 Prime numbers 16 The fundamental theorem of arithmetic 17 18 The series of reciprocals of the primes The Euclidean algorithm 19 The greatest common divisor of more than two numbers Exercises for Chapter 21 Chapter 20 Arithmetical 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 Theorem Functions and Dirichlet Multiplication Introduction 24 The Mobius function p(n) 24 The Euler totient function q(n) 25 A relation connecting rp and p 26 A product formula for q(n) 27 The Dirichlet product of arithmetical functions 29 Dirichlet inverses and the Mobius inversion formula 30 The Mangoldt function A(n) 32 Multiplicative functions 33 35 Multiplicative functions and Dirichlet multiplication 36 The inverse of a completely multiplicative function vii 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 Liouville’s function l(n) 37 The divisor functions e,(n) 38 Generalized convolutions 39 Forma1 power series 41 The Bell series of an arithmetical function 42 Bell series and Dirichlet multiplication 44 Derivatives of arithmetical functions 45 The Selberg identity 46 Exercises for Chapter 46 Chapter Averages of Arithmetical 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 Functions Introduction 52 53 The big oh notation Asymptotic equality of functions Euler’s summation formula 54 Some elementary asymptotic formulas 55 The average order of d(n) 57 The average order of the divisor functions a,(n) 60 The average order of q(n) 61 An application to the distribution of lattice points visible from the origin The average order of p(n) and of A(n) 64 The partial sums of a Dirichlet product 65 Applications to p(n) and A(n) 66 69 Another identity for the partial sums of a Dirichlet product Exercises for Chapter 70 Chapter Some Elementary Theorems on the Distribution Numbers 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 Introduction 74 Chebyshev’s functions t&x) and 9(x) 75 Relations connecting 8(x) and n(x) 76 Some equivalent forms of the prime number theorem 79 Inequalities for n(n) and p, 8.2 Shapiro’s Tauberian theorem 85 Applications of Shapiro’s theorem 88 An asymptotic formula for the: partial sums cPsx (l/p) 89 The partial sums of the Mobius function 91 Brief sketch of an elementary proof of the prime number theorem Selberg’s asymptotic formula 99 Exercises for Chapter 101 Chapter Congruences 5.1 5.2 5.3 Vlll of Prime Definition and basic properties of congruences Residue classes and complete residue systems Linear congruences 110 106 JO9 98 62 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 213 Reduced residue systems and the Euler-Fermat theorem 114 Polynomial congruences module p Lagrange’s theorem 115 Applications of Lagrange’s theorem Simultaneous linear congruences The Chinese remainder theorem 118 Applications of the Chinese remainder theorem Polynomial congruences with prime power moduli 120 123 The principle of cross-classification A decomposition property of reduced residue systems 125 Exercises,fbr Chapter 126 117 Chapter Finite Abelian 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 Groups and Their Characters Definitions 129 130 Examples of groups and subgroups Elementary properties of groups 130 Construction of subgroups 131 Characters of finite abelian groups 133 The character group 135 The orthogonality relations for characters 136 Dirichlet characters 137 Sums involving Dirichlet characters 140 The nonvanishing of L( 1, x) for real nonprincipal Exercises,for Chapter 143 x 141 Chapter Dirichlet’s 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 Theorem on Primes in Arithmetic Progressions Introduction 146 Dirichlet’s theorem for primes of the form 4n - and 4n + 148 The plan of the proof of Dirichlet’s theorem Proof of Lemma 7.4 150 Proof of Lemma 7.5 151 Proof of Lemma 7.6 152 Proof of Lemma 7.8 153 Proof of Lemma 7.7 153 Distribution of primes in arithmetic progressions 154 Exercises for Chapter 155 147 Chapter Periodic Arithmetical 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Functions and Gauss Sums Functions periodic modulo k 157 Existence of finite Fourier series for periodic arithmetical functions Ramanujan’s sum and generalizations 160 Multiplicative properties of the sums S&I) 162 Gauss sums associated with Dirichlet characters 165 Dirichlet characters with nonvanishing Gauss sums 166 Induced moduli and primitive characters 167 158 ix 8.8 8.9 8.10 8.11 8.12 Further properties of induced moduli 168 The conductor of a character 271 Primitive characters and separable Gauss sums 171 The finite Fourier series of the Dirichlet characters I72 P6lya’s inequality for the partial sums of primitive characters Exercises for Chapter 175 173 Chapter Quadratic 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 Residues and the Quadratic Quadratic residues 178 Legendre’s symbol and its properties 179 Evaluation of (- Jp) and (2 Ip) 182 Gauss’ lemma 182 The quadratic reciprocity law 185 Applications of the reciprocity law 186 The Jacobi symbol 187 Applications to Diophantine equations 190 Gauss sums and the quadratic reciprocity law The reciprocity law for quadratic Gauss sums Another proof of the quadratic reciprocity law Exercises for Chapter 201 Reciprocity Law 192 195 200 Chapter 10 Primitive 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 Roots The exponent of a number mod m Primitive roots 204 Primitive roots and reduced residue systems 205 The nonexistence of primitive roots mod 2” for a 206 The existence of primitive roois mod p for odd primes p 206 Primitive roots and quadratic residues 208 The existence of primitive roots mod p” 208 The existence of primitive roots mod 2p” 210 The nonexistence of primitive roots in the remaining cases 211 The number of primitive roots mod m 212 The index calculus 213 Primitive roots and Dirichlet characters 218 Real-valued Dirichlet characters mod p’ 220 Primitive Dirichlet characters mod p” 221 Exercises for Chapter 10 222 Chapter 11 Dirichlet 11.1 11.2 11.3 X Series and Euler Products Introduction 224 The half-plane of absolute convergence of a Dirichlet The function defined by a Dirichlet series 226 series 225 14: Partitions Equation (25) can also be written as follows: o(n) - a(n - 1) - o(n - 2) + o(n - 5) + a(n - 7) - (- l)“-‘w(m) = (-l)“-‘o(-m) i0 The sum on the left terminates when the term a(k) has k < when n = and n = this gives the relations if n = w(m), ifn = o(-mm), otherwise To illustrate, o(6) = o(5) + o(4) - a(l), o(7) = a(6) + o(5) - o(2) - 14.11 The partition identities of Ramanujan By examining MacMahon’s table of the partition function, Ramanujan was led to the discovery of some striking divisibility properties of p(n) For example, he proved that (26) p(5m + 4) = (mod 5), (27) p(7m + 5) E (mod 7), (28) ~(1 lm + 6) E (mod 11) In connection with these discoveries he also stated without remarkable identities, (29) $p(5m proof two + 4)x” = $$, and (30) m$op(7m + 5)xm = 7’s + 49x @$J, where p(x) = fi (1 - x”) VI=1 Since the functions on the right of (29) and (30) have power series expansions with integer coefficients, Ramanujan’s identities immediately imply the congruences (26) and (27) Proofs of (29) and (30), based on the theory of modular functions, were found by Darling, Mordell, Rademacher, Zuckerman, and others Further proofs, independent of the theory of modular functions, were given by Kruyswijk [36] and later by Kolberg Kolberg’s method gives not only the Ramanujan identities but many new ones Kruyswijk’s proof of (29) is outlined in Exercises 1l-l 324 Exercises for Chapter Exercises for Chapter 14 14 Let A denote a nonempty set of positive integers (a) Prove that the product is the generating function of the number of partitions of n into parts belonging to the set A (b) Describe the partition function generated by the product In particular, describe r-IL1 (1 + xrn) the partition function generated by the finite product If 1x < prove that &l + xm) = iy (1 - xzm- 1)) 1, m=l and deduce that the number of partitions number of partitions of n into odd parts For complex of n into unequal parts is equal to the x and z with 1x < 1, let j-(x, z) = fi (1 - xmz) Ill=1 (a) Prove that for each fixed z the product is an analytic function of x in the disk 1x < 1, and that for each fixed x with 1x < the product is an entire function of z (b) Define the numbers a,(x) by the equation f(x, z) = f u,(X)Zn, n=o Show thatf(x, z) = (1 - xz)f(x, satisfy the recursion formula zx) and use this to prove that the coefficients a,(x) = a,(x)x” - a,-,(x)x” (c) From part (b) deduce that a,(x) = (- l)“~“(“+~)~~/P~(x), P”(X) = l-&l ,=, This proves the following identity where - x’) for 1x < and arbitrary z: 325 14: Partitions Use a method we have analogous to that of Exercise to prove that if lx < and lzl < *g,u -xmz)-l =“j&” where P,(x) = n:= (1 - xr) If x # let Q,,(x) = and for n > define (a) Derive the following finite identities F m=l of Shanks: X*0- 1)/Z= El ~Q.(x) XS(2n+ s=o QsC4 1) ’ 2n+l (b) Use Shanks’ identities to deduce Gauss’ triangular-number theorem: for 1x1 < f m=l The following identity is valid for 1x < 1: X*(*+ U/2 = fJ (1 + Xn- I)(1 - x2y, f *=-LX n=1 (a) Derive this from the identities in Exercises and 5(b) (b) Derive this from Jacobi’s triple product identity Prove that the following triple product identity: (a) “fil(l - ,$“)(l (b) fi(1 - x5”)(1 - identities, valid for 1x < 1, are consequences x5n-4) = - f+l)(l x5n-2)(1 X5n-3) = n=l of Jacobi’s f (-l)mX*(5*+3)12 *= m $ (-l)*X*(~*+1K2~ *=-LX Prove that the recursion formula v(n) = i 4kMn - 4, k=l obtained in Section 14.10, can be put in the form q(n) = i m=l ksnlm mp(n - km) Suppose that each positive integer k is written in g(k) different colors, where g(k) is a positive integer Let p,(n) denote the number of partitions of n in which each part k appears in at most g(k) different colors When g(k) = for all k this is the 326 Exercises for Chapter unrestricted partition function p(n) Find p,(n) and prove that there is an arithmetical an infinite product functionf(depending 14 which generates on g) such that v,(n) = i f(k)p,(n - k=l 10 Refer to Section 14.10 for notation By solving in (22) prove that if 1x < we have “p - y-m/n the first-order differential equation = exp{[Iq)dt}, where ff(x) = fXdxk and f2k) = C f(4 k=l Deduce that fj(1 where p(n) is the Mobius The following a proof of Ramanujan’s + 4)x” = $f$, of Kruyswijk for 1x1 < 1, function exercises outline m$~(5m by a method - x”)~(“)~” = eex partition identity where q(x) = fi (1 - x”), n=1 not requiring the theory of modular functions, 11 (a) Let E = eZniik where k and show that for all x we have *fil(l (b) More generally, - X&h) = - xk if (n, k) = d prove that *f!(l - X&“h) = (1 - Xk’d)d, and deduce that if(n,k)= *fil(l - Xne2nWk) - {il-z’)k 12 (a) Use Exercise l(b) to prove that for prime “I1 *fil(l 1, ifkln q and Ix < we have - x”eZninhiq)= s (b) Deduce the identity mfj,p(rn)xm = $$ *fJl “fir(l - xneZainh’5) 327 14: Partitions 13 If q is prime and if I < q, a power series of the form is said to be of type r mod q (a) Use Euler’s pentagonal number power series, theorem q(x) =: fi(l “=I to show that q(x) is a sum of three - x”) = I, + I, + I,, where I, denotes a power series of type k mod (b) Let tl = ezni15 and show that x”a”h) = fi(r, + I,ah+ I,CrZh) h=l (c) Use Exercise 12(b) to show that fop(5m + 4)X5m+4 = v, sj, where V, is the power series of type mod obtained 14 (a) Use Theorem power series, from the product in part(b) 14.7 to show that the cube of Euler’s product is the sum of three q’(x)3 = w, + w, + w,, where W, denotes a power series of type k mod (b) Use the identity W, + W, + W, = (I, + I, + 1J3 to show that the power series in Exercise 13(a) satisfy the relation I,12 = -I,Z (c) Prove that I, = -x(p(xz5) 15 Observe that the product flz= r (I, + 1,~~ + 1,~‘~) is a homogeneous polynomial in I,, I,, I, of degree 4, so the terms contributing to series of type mod come from the terms 114, 1,1,212 and I,’ I,‘ (a) Use Exercise 14(c) to show that there exists a constant c such that v, = cI,4, where V, is the power series in Exercise 13(c), and deduce that m;/hl + 4)xSm+4 = cx4$g (b) Prove that c = and deduce Ramanujan’s m$(5m 328 identity + 4)xm = ‘PO5 (Pc46 Bibliography MR denotes reference to Mathematical Reviews Apostol, Tom M (1970) Euler’s q-function Math Sot., 24: 482485; MR 41, # 1661 Apostol, Tom M (1974) Mathematical Wesley Publishing Co and separable Gauss sums Proc Amer Analysis, 2nd ed Reading, Mass : Addison- Ayoub, Raymond G (1963) An Introduction to the Analytic Theory of’ Numbers Mathematical Surveys, No 10 Providence, R I.: American Mathematical Society Bell, E T (1915) An arithmetical theory of certain numerical functions qf Washington Publ in Math and Phys Sci., No 1, Vol 1: l-44 Borozdkin, K G (1956) K voprosu o postoyanni I M Vinogradova tretqqo vsesoiuzqao matematifesko,qo siezda, Vol I, Moskva [Russian] University Trudy BuhStab, A A (1965) New results in the investigation of the Goldbach-Euler problem and the problem of prime pairs [Russian] Dokl Akad Nauk SSSR, 162: 735-738; MR 31, #2226 [English translation: (1965) Soviet Math Dokl 6: 729-732.1 Chandrasekharan, Die Grundlehren Springer-Verlag Komaravolu (1968) Introduction to Analytic der Mathematischen Wissenschaften, Band Number Theory 148 New York: Chandrasekharan, der Mathematischen Komaravolu (1970) Arithmetical Functions Die Grundlehren Wissenschaften, Band 167 New York Springer-Verlag Chebyshev, P L Sur la fonction qui determine la totalite des nombres premiers inferieurs a une hmite donee (a) (1851) Mem AC SC St PPtersbour,q, 6: 141-157 (b) (1852) hna de Math (1) 17: 341-365 [Oeuvres, I: 2748.1 10 Chen, Jing-run (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes Kexue Toqbao (Foreign Lang Ed.), 17: 3855386; MR 34, #7483 329 Bibliography 11 Clarkson, James A (1966) On the series of prime Sot., 17: 541; MR 32, # 5513 12 Davenport, Harold (1967) Multiplicative Mathematics, No Chicago: Markham reciprocals Number Theory Publishing Co 13 Dickson, Leonard Eugene (1919) History Washington, D C.: Carnegie Institution Publishing Co., New York, 1966 Proc Amer Math Lectures of the Theory of Numbers of Washington Reprinted 14 Dickson, Leonard Eugene (1’930) Studies The University of Chicago Press in the Theory in Advanced (3 volumes) by Chelsea of Numbers Chicago: 15 Dirichlet, P G Lejeune (183’7) Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschlaftichen Factor sind, unendliche viele Primzahlen enthglt Abhand Ak Wiss Berlin: 45-81 [Werke, 1: 315-342.1 16 Dirichlet, P G Lejeune (1840) Ueber eine Eigenschaft Bericht Ak Wiss Berlin: 49-52 [Werke, 1: 497-502.1 der quadratischen Formen 17 Edwards, H M (1974) Riemann’s Zeta Function New York and London: Academic Press 18 Ellison, W J (1971) Waring’s problem Amer Math Monthly, 78: 10-36 19 ErdGs, Paul (1949) On a new method in elementary number theory which leads to an elementary proof of the prime number theorem Proc Nut Acad Sci U.S.A., 35: 374-384: MR 10.595 20 Euler, Leonhard (1737) Variae observationes circa series infinitas Commentarii Academiue Scientiurum Imperialis Petropolitunae, 9: 16&l% [Opera Omniu (l), 14; 216244.1 21 Euler, Leonhard (1748) lntroductio in Anulysin Injinitorum, Vol Lausanne: Bousquet [Opera Omniu (l), 8.‘] 22 Franklin, F (1881) Sur le dCveloppement du produit infini (1 - x)(1 - x2) (1 - x3)( - x4) Comptes Rendus Acad Sci (Paris), 92: 448-450 23 Gauss, C F (1801) Disquisii!iones Arithmeticae Lipsiae [English translation: Arthur A Clarke (1966) New Haven : Yale University Press 24 Gauss, C F (1849) Letter to Encke, dated 24 December [We&e, Vol 25 Gerstenhaber, Murray (1963) The 152nd proof of the law of quadratic Amer Math Monthly, 70: 397-398; MR 27, # 100 26 Goldbach, C (1742) Letter to Euler, dated June 27 Grosswald, Emil (1966) Topics from the Theory of Numbers New Macmillan Co 28 Hadamard, J (1896) Sur la distribution des ztros de la fonction c(s) quences arithmttiques Bull Sot Math France, 24: 199-220 29 Hagis, Peter, Jr (1973) A lower bound Comp., 27: 951-953; MR 48, #3854 II, 444447.1 reciprocity York: The et ses cons& for the set of odd perfect numbers Math 30 Hardy, G H (1940) Rumunujun Twelve Lectures on Subjects Suyyested by His Lsfe and Work Cambridge : The University Press 31 Hardy, G H and Wright, E M (1960) An Introduction to the Theory of Numbers, 4th ed Oxford: Clarendon Press 32 Hemer, Ove (1954) Notes on the Diophantine equation y2 - k = x3 Ark Mat., 3: 67-77; MR 15, 776 33 Ingham, A E (1932) The Distribution of Prime Numbers Cambridge Tracts in Mathematics and Mathematical Physics, No 30 Cambridge: The University Press 330 Bibliography 34 Jacobi, C G J (1829) Fundamenta [Gesammelte Werke, Band I, 49-239.1 Nova Theoriae Functionum Ellipticarum 35 Kolesnik, G A (1969) An improvement of the remainder term in the divisor problem (Russian) Mat Zumetki, 6: 545-554; MR 41, # 1659 [English translation: (1969) Math Notes, 6: 784791.1 36 Kruyswijk D (1950) On some well-known properties of the partition function p(n) and Euler’s infinite product Nieuw Arch Wisk., (2) 23: 97-107; MR 11, 715 37 Landau, E (1909) Handbuch der Lehre von der Verteilung der Primzahlen Leipzig : Teubner Reprinted by Chelsea, 1953 38 Landau, E (1927) Vorlesungen iiber Zahlentheorie (3 volumes) Leipzig: Hirzel Reprinted by Chelsea, 1947 39 Leech, J (1957) Note on the distribution of prime numbers J London Math Sot., 32: 56-58; MR 18,642 40 Legendre, A M (1798) Essai sur la Theorie des Nombres Paris : Duprat 41 Lehmer, D H (1959) On the exact number of primes less than a given limit Illinois J Math., 3: 381-388; MR 21, #5613 42 Lehmer, D H (1936) On a conjecture of Ramanujan J London Math Sot., 11: 114118 43 Lehmer, D N (1914) List of prime numbers from to 10, 006, 721 Washington, D.C : Carnegie Institution of Washington, Publ No 165 44 LeVeque, W J (1956) Topics in Number Theory (2 volumes) Reading, Mass.: Addison-Wesley Publishing Co 45 LeVeque, W J (1974) Reviews in Number Theory (6 volumes) Providence, RI: American Mathematical Society 46 Levinson, N (1969) A motivated account of an elementary proof of the prime number theorem Amer Math Monthly, 76: 225-245; MR 39, #2712 47 Levinson, Norman (1974) More than one third of zeros of Riemann’s are on o = l/2 AdvancesMath., 13: 383436 zeta-function 48 van Lint, Jacobus Hendricus (1974) Combinatorial Theory Seminar (Eindhoven University of Technology), Lecture Notes in Mathematics 382 Springer-Verlag, Chapter 49 Littlewood, J E (1914) Sur la distribution des nombres premiers Comptes Rendus Acad Sci (Paris), 158: 186991872 50 Mills, W H (1947) A prime-representing function Bull Amer Math Sot., 53: 604; MR 8,567 51 Nevanlinna,’ V (1962) Uber den elementaren Beweis des Primzahlsatzes Sot Sci Fenn Comment Phys-Math, 27 No 3, pp.; MR 26, #2416 52 Niven, I and Zuckerman, H S (1972) An Introduction to the Theory of Numbers, 3rd ed New York: John Wiley and Sons, Inc 53 Prachar, Karl (1957) Primzahlverteilung Die Grundlehren der Mathematischen Wissenschaften, Band 91 Berlin-Gottingen-Heidelberg : Springer-Verlag 54 Rademacher, Hans (1937) On the partition functionp(n) Proc London Math Sot., 43: 241-254 55 Rademacher, Hans (1964) Lectures on Elementary Number Theory New York: Blaisdell Publishing Co 56 Rademacher, Hans (1973) Topics in Analytic Number Theory Die Grundlehren Mathematischen Wissenschaften, Band 169 New York-Heidelberg-Berlin: Springer-Verlag der 331 Bibliography 57 Renyi, A (1948) On the representation of an even number as the sum of a single prime and a single almost-prime number (Russian) Zzv &ad Nauk SSSR Ser Mat., 12: 57-78; MR 9,413 [English translation: (1962) Amer Math Sot Transl 19 (2): 299-321.1 58 Riemann, B (1859) Uber die Anzahl der Primzahlen unter einer gegebener Griisse Monatsber Akad Berlin, 671-680 59 Robinson, R M (1958) A report on primes of the form k ‘2” + and on factors of Fermat numbers Proc Amer Math Sot., 9: 6733681; MR 20, #3097 60 Rosser, J Barkley, and Schoenfeld, Lowell (1962) Approximate formulas for some functions of prime number theory Illinois J Math., 6: 69-94; MR 25, # 1139 61 Schnirelmann, L (1930) On additive properties of numbers (Russian) Izc Donskowo Politechn Inst (Nowotscherkask), 14 (2-3): 3328 62 Selberg, Atle (1949) An elementary proof of the prime number theorem Ann of Math., 50: 305-313; MR 10, 595 63 Shanks, Daniel (1951) A short proof of an identity of Euler Proc Amer Math Sot 2: 747-149; MR 13, 321 64 Shapiro, Harold N (1950) On the number of primes less than or equal x Proc Amer Math Sot., I: 346348; MR 12, 80 65 Shapiro, Harold N (1952) On primes in arithmetic progression II Ann qf Math 52: 231-243: MR 12 81 66 Shen, Mok-Kong (1964) On checking the Goldbach conjecture Nordisk Tidskr Infbrmations-Behandling, 4: 243-245 ; MR 30, # 305 67 Sierpidski, Waclaw (1964) Elementary Theory of Numbers Translated from Polish by A Hulanicki Monografie Matematyczne, Tom 42 Warsaw: Panstwowe Wzdawnictwo Naukowe 68 Tatuzawa, Tikao, and Iseki Kaneshiro (1951) On Selberg’s elementary proof of the prime number theorem Proc Japan Acad., 27: 340-342; MR 13,725 69 Titchmarsh, E C (1951) The Theory of the Riemann Zeta Function Oxford: Clarendon Press 70 Uspensky, J V., and Heaslett, M A (1939) Elementary Number Theory New York: McGraw-Hill Book Co 71 Vallee Poussin, Ch de la (1896) Recherches analytiques sur la theorie des nombres premiers Ann Sot Sci Bruxelles, 20, : 183-256, 281-297 72 Vinogradov, A I (1965) The density hypothesis for Dirichlet L-series (Russian) Izv Akad Nauk SSSR, Ser Math 29: 903-934; MR 33, #5579 [Correction: (1966) ibid., 30: 719-720; MR 33, #2607.] 73 Vinogradov, I M (1937) The representation of an odd number as the sum of three primes (Russian.) Dokl Akad Nuuk SSSR, 16: 139-142 74 Vinogradov, I M (1954) Elements of Number Theory Translated by S Kravetz New York : Dover Publications 15 Walfisz, A (1963) Weylsche Exponentialsummen in der neueren Zahlentheorie Mathematische Forschungsberichte, XV, V E B Deutscher Verlag der Wissenschaften, Berlin 76 Williams, H C., and Zarnke, 2A3” + and 2A3” - Math C R (1972) Some prime numbers Comp 26: 995-998; MR 41, #3299 of the form 77 Wrathall, Claude P (1964) New factors of Fermat numbers Math Comp., 18: 324-325; MR 29, # 1167 78 Yin, Wen-lin (1956) Note on the representation of large integers as sums of primes Bull Acad Polon Sci Cl 111, 4: 793-795; MR 19 16 332 Index of Special Symbols d In, d Y n, (4 b), h, 4, [a, bl, An), cp(n), f * 99 [I divides (does not divide), 14 greatest common divisor (gcd),l5,21 least common multiple (lcm), 22 Mobius function, 24 Euler totient, 25 Dirichlet convolution, 29 Z(n) = i , identity function, 30 f-5 Dirichlet inverse, 30 unit function, 31 Mangoldt function, 32 Liouville function, 37 divisor functions, 38 generalized convolution, 39 Bell series off modulo p, 43 derivative, 45 Euler’s constant, 53 big oh notation, 53 asymptotic equality, 53 Riemann zeta function, 55 number of primes IX, 74 Chebyshev +-function, 75 u(n) = 1, A(n), W, o,(n), 44, d(n), a F, fpM f ‘(4 = f (n)logn, C, 0, i;s;, 4x), $64 333 Index of special symbols %4> M(x), 0, a = b (mod m), ^ a, I a, x(n), L(L xl, L’U, x), 44, G(n, xl, W ; 4, nRp, nb, b IPX (n I PI, evmk4 ind, a, Lb, xl, 0a, fJ ri;,, as, 4, F(x, 4, 8k4 4, m4 p(n), dn),4 334 - 4, Chebyshev &function, 75 partial sums of Mobius function, 91 little oh notation, 94 congruence, 106 residue class a module m, 109 reciprocal of a modulo m, 111 Dirichlet character, 138 sum of series C x(n)/n, 141 sum of series -c X(n)log n/n, 148 Ramanujan sum, 160 Gauss sum associated with x, 165 quadratic Gauss sum, 177 quadratic residue (nonresidue) mod p, 178 Legendre symbol, 179 Jacobi symbol, 188 exponent of a module m, 204 index of a to base g, 213 Dirichlet L-function, 224 abscissa of absolute convergence, 225 abscissa of convergence, 233 gamma function, 250 Hurwitz zeta function, 251 periodic zeta function, 257 Bernoulli polynomials, (numbers), 264 periodic Bernoulli functions, 267 partition function, 307 pentagonal numbers, 311 Index Abel, Niels Henrik, 77 Abelian group, 129 Abel’s identity, 77 Abscissa, of absolute convergence, 225 of convergence 233 Additive number theory, 304 Algorithm, division, 19 Euclidean, 20 Analytic continuation, of Dirichlet L-functions, 255 of Hurwitz zeta function, 254 of Riemann zeta function, 255 Apostol, Tom M., 329 Arithmetic, fundamental theorem of, 17 Arithmetical function, 24 Asymptotic equality, 53 Average (arithmetic mean) of an arithmetical function, 52 Average order, 52 Ayoub, Raymond G., 329 Bell, Eric Temple, 29, 42, 329 Bell series, 43 Bernoulli, numbers, 265 periodic functions, 267 polynomials, 264 Binomial congruence, 214 Borozdkin, K G., 10, 329 BuhStab, A A., 11,329 Cauchy, Augustin-Louis, 44, 144, 198 Cauchy product, 44 Chandrasekharan, Komaravolu, 329 Character, Dirichlet, 138 of an abelian group, 133 primitive, 168 principal, 138 Chebyshev, Pafnuti Liwowich, 75 Chebyshev function 9(x), 75 Chebvshev function tilx) 75 Chen: Jing-run, I 1, 3b4,329 Chinese remainder theorem, I 17 Clarkson, James A., 18,330 Classes of residues, 109 Clausen, Thomas, 275 Common divisor, 14 Commutative group, 129 Complete residue system, 110 Completely multiplicative function, 33 Conductor of a character, 17 Congruence, 106 Convolution Dirichlet 29 generalized, 39 van der Corput, J G., 59 Critical line 293 Critical strip, 293 Cross-classification principle, 123 Cyclic group, 131 Darling, H B C., 324 Davenport, Harold, 330 Decomposition property of reduced residue systems, 125 Derivative of arithmetical functions, 45 Dickson, Leonard Eugene, 12, 330 Diophantine equation, 5, 190 Dirichlet, Peter Gustav Lejeune, 5, 7, 29, 53, 138 146,224 Dirichlet, character I 138 convolution (product) 29 divisor problem, 59 estimate for d(n), 53, 57 inverse, 30 L-function, 224 series, 224 theorem on primes in arithmetic progressions, 7, 146, 154 Disquisitiones arithmeticae, Divisibilitv, 14 Division algorithm, 19 Divisor, 14 Divisor function e.(n), 38 Edwards, H M., 330 Elementary proof of prime number theorem, 9.98 Ellison, W J., 306, 330 Erdiis, Paul, 9, 330 Euclidean algorithm, 20 Euclid’s lemma, 16 Euler, Leonhard, 4, 5, 7, 9, 19, 25, 53, 54, 113, 180,185,230,308, 312,315 Euler-Fermat theorem, 113 Euler product, 230 Euler’s constant, 53, 250 Euler’s criterion, 180 335 Index Euler’s pentagonal-number theorem, 12 Euler’s summation formula, 54 Euler totient function (p(a) 25 Evaluation, of ( - 11p), 181 of(2lp), 181 of [( -n, a), 264 of 1(2n), 266 of L(0, x), 268 Exponent of a module m, 204 Exponential congruence, 215 Factor, 14 Fermat, Pierre de, 5, 7, 11, 113, 114 Fermat conjecture, I I Fermat prime, 7, Fermat theorem (little), 114 Finite Fourier expansion, 160 Formal power series, 41 Fourier coefficient, 160 Franklin, Fabian, 13, 330 Function, arithmetical, 24 Bernoulli neriodic B.(x) 267 Chebyshev 9(x), 75 “’ Chebyshev I/I(Y), 75 completely multiplicative, 33 Dirichlet L(s, x), 224 divisor d(n), u,(n), 38 Euler totient cp(n), 25 Hurwitz zeta ((s, a), 249 Liouville I(n), 37 Mangoldt A(n) 32 Mobius &I), 24 periodic zeta F(x, s), 257 Riemann zeta c(s), 9, 249 K(n), 247 M(x), v(n), 247 n(x), 8.74 Ii/,(x), 278 Functional equation, for F(s), 250 for L(s, x), 263 for i(s), 259 for i(s, h/k), 261 Fundamental theorem of arithmetic, 17 Gamma function, 250 Gauss, Carl Friedrich 5, 7, 8, 106, 165, 177, 182, 185, 306, 326 Gauss sum, associated with x, 165 quadratic, 177, 306 Gauss’ lemma, 182 Gauss’ triangular-number theorem, 326 Generating function, 308 Geometric sum, 157, 158 Gerstenhaber, Murray, 186, 330 Goldbach, C., 6,9, 304 Goldbach conjecture, 9, 304 Greatest common divisor, 15,20, 21 Greatest integer symbol, 8,25, 54, 72 Grosswald, Emil, 330 336 Group, definition abelian 129 cyclic, I3 I Group character of 129 133 Hadamard, Jacques, 9,74, 330 Hagis, Peter, Jr., 5, 330 Half-plane, of absolute convergence, 225 of convergence, 233 Hardy, Godfrey Harold, 59,293, 305, 330 Hemer, Ove, 330 Hilbert, David, 293 Hurwitz, Adolf, 249 Hurwitz formula for {(s, u), 257 Hurwitz zeta function [(s, a), 249, 251, 253, 255 Identity element 30, 129 Identity function I(n), 30 Index, 213 Index calculus, 214 Indices (table of),216, 217 Induced modulus, 167 Induction, principle of, 13 Infinitude of primes, 16, 19 Inequalities, for 1[(s, a) 1, 270 for 1i(s) 1, 270, 287, 291 for I Us, x) I, 272 for n(n), 82 for nth prime p 84 for d(n), 294 for q(n), 298 for p(n), 316, 318 Ingham, A E., 330 Inverse, Dirichlet, 30 of completely multiplicative function, Inversion formula, Mobius, 32 generalized, 40 Iseki, Kaneshiro, 99, 332 Jacobi, Jacobi Jacobi Jordan 36 Carl Gustav Jacob, 187,305, 313, 319 symbol (n I P), 188 triple product identity, 319 totient Jk(n), 48 Kloosterman, H D., 176 Kloosterman sum, 176 Kolberg, Oddmund, 324 Kolesnik G A 59 Kruyswijk, D., 324, 327, 331 Lagrange, Joseph Louis, 5, 115, 144, 158 Lagrange interpolation formula, 158 Lagrange’s theorem on polynomial congruences, 115 Landau, Edmund, 59,237,248,301,331 Landau’s theorem, 237,248 Lattice points, 57, 62 visibility of, 62 Index Law of quadratic reciprocity 185 189, 193.200 Least common multiple, 22 Leech John 10.331 Legendre, Adrien-Marie, 5, 67, 179, 185, 331 Legendre’s identity, 67 Legendre symbol (n Ip), 179 Lehmer, Derrick Henry, 6, 293, 316, 331 Lehmer, Derrick Norman, 6, 331 Lemma of Gauss, 182 LeVeque, William Judson, 2, 191, 33 Levinson, Norman, 293, 33 L-function L(s, x), 224 Linear congruence, 111, 112, 114,214 van Lint, Jacobus Hendricus, 318, 331 Liouville, Joseph, 37 Liouville function A(n), 37 Little Fermat theorem, I14 Littlewood, John Edensor, 10, 305, 331 Logarithmic integral Li(x), 102 Lucas, Edouard, 275 MacMahon, Percy A., 316 von Mangoldt, H., 32 von Mangoldt function A(n), 32 Mean value formulas for Dirichlet series, 240 Mersenne, P., Mersenne numbers, Mertens, Franz, 91 Mertens’ conjecture, Mills, W H., 8, 331 Mobius, Augustus Ferdinand, 24 Mobius function p(n), 24 Mobius function pLk(n) of order k, 50 Mobius inversion formula, 32 product form, 47 generalized, 40 Mordell, Louis Joel 305 306 Multiplication, Dirichlet 29 of residue classes, 138 Multiplicative function, 33 Multiplicative number theory, 304 Nevanlinna, Veikko, 331 Niven, Ivan, 331 Nonresidue, 178 Number-theoretic function, 24 0, big oh notation, 53 o, little oh notation, 94 Order of a group, 130 Orthogonality relation, for group characters, 137 for Dirichlet characters, 140 Partition, 304 Partition function p(n), 307 Pentagonal numbers, 2, 5, 311 Pentagonal-number theorem, 312 Perfect numbers, Periodic arithmetical function, 157 Periodic zeta function, 257 Polya, G., 173, 299 Polya inequality for character sums 173 176 299 Polygonal numbers, 2, Polynomial congruence, 115 Prachar, Karl, 331 Prime number theorem, 9,65,74, 79,92,94, 98,278,289 Primes, 2, 16, contained in a factorial, 67 in arithmetic progressions, 7, 146, 154 Fermat, infinitude of, 16, 19 Mersenne, Primitive character, 168 Primitive root, 204 Principal character, 134, 138 Product, of arithmetical functions, 29 of Dirichlet series, 228 Pythagorean triple, Quadratic, congruence, 178 Gauss sum, 177, 195 nonresidue, 178 reciprocity law, 185, 189, 193, 200 residue 178 Rademacher, Hans, 16, 33 Ramanuian Srinivasa 160 305 324 328 Ramanujan’partition identities, 324,‘328 Ramanujan sum, 160, 176 Reciprocity law, for Jacobi symbols, 189 for Legendre symbols, 185, 193,200 for quadratic Gauss sums, 200 Reduced fraction, 21 Relatively prime, 15, 21 in pairs, 21 Rinyi, Alfred, 10, 332 Residue, quadratic, 178 Residue class, 109 Residue system, complete, 110 reduced, 113, 125 Riemann, Georg Friedrich Bernhard, 9,225 293,332 Riemann hypothesis,,293, 301 Riemann-Lebesgue lemma, 279 Riemann-Stieltjes integral, 77 Riemann zeta function, 249 Ring of formal power series, 42 Robinson, Raphael M., 7, 332 Rosser, J Barkley, 293, 332 Schnirelmann, L., 10, 332 Selberg Atle, 46 100 332 Selberg asymptotic formula, 100, 103, 104 Selberg identity, 46 Separable Gauss sums, 165, 171 Shanks, Daniel, 312, 326 332 331 Index Shapiro, Harold N., 85, 146, 332 Shapiro’s Tauberian theorem, 85 Shen, Mok-Kong, 9,332 Sierpinski, Waclaw, 11, 148, 332 Smallest primitive roots (table of), 213 Squarefree, 21 von Staudt, Karl Georg Christian, 275 Subgroup, 130 Summation formula of Euler, 54 Symbol, Jacobi (nip), 188 Legendre (n 1p) 179 System of residues, complete, 110 reduced 113 Tatuzawa, Tikao, 99, 332 Tauberian theorem, 85 Theta function, 306 Titchmarsh, Edward Charles, 301, 332 Totient function cp(n), 25 Triangular numbers, 2, 326 Triangular-number theorem, 326 Trivial zeros of i(s), 259 Twin primes, Unique factorization theorem, 17 Uspensky, J V 332 338 Vallee-Poussin, C Vinogradov, A I., Vinogradov, I M., Visibility of lattice Voronoi, G., 59 J de la, 9, 74, 332 11, 332 10,304,332 points, 62 Wahisz, Arnold, 91, 332 Waring Edward 306 Warinzs problem, 306 Williams, H C 6, 332 Wilson, John, 116 Wilson’s theorem, I16 Wrathall, Claude P., 7, 332 Wright, E M., 330 Wolstenholme’s theorem, I16 Yin, Wen-lin, IO, 332 Zarnke, C R., 6,332 Zero-free regions of c(s), 29 1, 292 Zeros, of L-function L(s, x), 274 of Riemann’s zeta function c(s), 259, 274, 293 Zeta function, Hurwitz, 249 periodic, 257 Riemann, 9.249 Zuckerman Herbert S 331 ...Tom M Apostol Introduction to Analytic Number Theory Springer-Verlag New York 1976 Heidelberg Berlin Tom M Apostol Professor of Mathematics California... Classification 10-01, 1OAXX Library of Congress (1976) Cataloging in Publication Data Apostol, Tom M Introduction to analytic number theory (Undergraduate texts in mathematics) ” Evolved from a course (Mathematics... three-volume History of the Theory of Numbers [13], and LeVeque’s six-volume Reviews in Number Theory [45] Dickson’s History gives an encyclopedic account of the entire literature of number theory up