Modular functions and dirichlet series in number theory, tom m apostol

Graduate Texts in Mathematics 41 EditOl·ial Board s Axler F.W Gehring Springer Science+Business Medi~ LLC К.А Ribet Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 TAKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra VoU ZARISKI/SAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed 35 ALExANDERIWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory J 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRA VERlW ATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELL/Fox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol l 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed (continued after index) Tom M Apostol Modular Functions and Dirichlet Series in Number Theory Second Edition With 25 Illustrations Springer Aposto! Department of Mathematics Ca!ifornia Institute of Techno!ogy Pasadena, СА 91125 USA Тот М Еl!i/щiа! В{){т! S Axler Department of Mathematics San Francisco State University San Francisco, СА 94132 U.S.A F W Gсhгiпg Dерагtmепt К А af Mathcmatics Uпivеl'sitу of Мiсhigап Апп АгЬог М! Ribet af Mathematics University of California at Berke1ey Berkeley, СА 94720 U.S.A Dерагtmепt 48109 U.S.A Mathematics Subject Classification (2000): 11-01, IIFXX Library of Congress Cataloging-in-Publication Data Apostol, Тот М Modular functions and Dirichlet series in number theory/Tom М Apostol.-2nd ed р cm.-{Graduate texts in mathematics; 41) Includes bibliographical references ISBN 978-1-4612-6978-6 ISBN 978-1-4612-0999-7 (eBook) DOI 10.1007/978-1-4612-0999-7 Number theory Functions, Elliptic Functions, Modular Series, Dirichlet Title 11 Series QA241.A62 1990 512'.7- 0, we write c dr - c cSr + d = - - + d = - - ' r r' hence, r + d) - I.( CS - i(dr - c) e -1TiI2 , -Ir and therefore, {-i(cSr + d)}112 {-ir}1/2 = e- 1TiI4{-i(dr in (9) together with Lemma 3, we obtain (7) If d < 0, we write cSr + d -dr c C)}112 Using this +c = r + d = - -r so that in this case we have - i(cSr + d) = - i( - dr -Ir + c) 1Ti12 e, and therefore, {-i(cSr + d)}112 {-irfl2 = e1Ti/4{-i(-dr + C)f/2 Using this in (9) together with Lemma 3, we obtain (8) Remark on the root of unity e(A) Dedekind's functional equation (I), with an unspecified 24th root of unity e(A), follows immediately by extracting 24th roots in t~e functional equation for ~(r) Much of the effort in this theory is directed at showing that the root of unity s(A) has the form given in (2) It is of interest to note that a simple argument due to Dedekind gives the following theorem: Theorem If (I) holds whenever A = (: !) E rand c ¥- 0, then s(A) = ex p{1Ti(a l ;Cd - f(d, c»)} for some rational number f(d, c) depending only on d and c 193 Supplement to Chapter PROOF Let +b aT AT= CT + d be two transformations in r I a'T + b' AT= CT+d and having the same denominator CT a'd - and ad-bc=1 = b'c + d Then I, so both pairs a, b and a', b' are solutions of the linear Diophantine equation xd - = yc Consequently, there is an integer n such that a' = a + nc, = b' b + nd Hence, A'T = (a + nC)T + (b + nd) = aT + b + n = AT + n CT+d cT+d Therefore, we have 7/(A'T) = 7/(AT + n) = e1Tin/127/(AT) = e1Tin/12e(A){ - i(CT + d)}1127/(T), because of (1) On the other hand, (I) also gives us 7/(A 'T) = e(A'){ -i(CT + d)}1127/(T) Comparing the two expressions for 7/(A 'T), we find e(A') n = (a ' - a)lc, so = exp ( 7Ti(a12c l e(A') - a») e1TinI12e(A) But e(A) , or ( ') 7TlQ e(A)I exp - 12c This shows that the product exp( - ( ) = exp 7TlQ - 12c e(A) ~~:)e(A) depends only on C and d There- fore, the same is true for the product exp ( - 7Ti(a + 12c d») e(A) This complex number has absolute value and can be written as exp( 194 7Ti(a + 12c d») e(A) = exp(-7Tif(d, c» Supplement to Chapter for some real number fed c) depending only on c and d Hence, e(A) = exp{ 7T{a I;C d - fed C))} Because e 24 = I, it follows that 12cf(d c) is an integer, so fed c) rational number IS a 195 Bibliography Apostol, Tom M Sets of values taken by Dirichlet's L-series Proc Sympos Pure Math., Vol VIII, 133-137 Amer Math Soc., Providence, R.I., 1965 MR 31 # 1229 Apostol, Tom M Calculus, Vol II, 2nd Edition John Wiley and Sons, Inc New York, 1969 Apostol, Tom M Mathematical Analysis, 2nd Edition Addison-Wesley Publishing Co., Reading, Mass., 1974 Apostol, Tom M Introduction to Analytic Number Theory Undergraduate Texts in Mathematics Springer-Verlag, New York, 1976 Atkin, A O L and O'Brien, J N Some properties of p(n) and c(n) modulo powers of 13 Trans Amer Math Soc 126 (1967),442-459 MR 35 #5390 Bohr, Harald Zur Theorie der allgemeinen Dirichletschen Reihen Math Ann 79 (1919), 136-156 Deligne, P La conjecture de Wei! I Inst haut Etud sci., Publ math 43 (1973), 273-307 (1974) Z 287, 14001 Erdos, P A note on Farey series Quart Math., Oxford Ser 14 (1943), 82-85 MR 5, 236b Ford, Lester R Fractions Amer Math Monthly 45 (1938), 586-601 10 Gantmacher, F R The Theory oj Matrices, Vol I Chelsea Pub! Co., New York, 1959 II Gunning, R C Lectures on Modular Forms Annals of Mathematics Studies, No 48 Princeton Univ Press, Princeton, New Jersey, 1962 MR 24 #A2664 12 Gupta, Hansraj An identity Res Bull Panjab Univ (N.S.) 15 (1964), 347-349 (1965) MR 32 #4070 13 Hardy, G H and Ramanujan, S Asymptotic formulae in combinatory analysis Proc London Math Soc (2) 17 (1918),75-115 14 Haselgrove, C B A disproof of a conjecture of P6lya Mathematika (1958), 141-145 MR 21 #3391 15 Hecke, E Uber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung Math Ann 112 (1936), 664 699 196 Bibliography 16 Hecke, E Ober Modulfunktionen und die Dirichlet Reihen mit Eulerscher Produktentwicklung I Math Ann JJ4 (1937), 1-28; II 316-351 17 Iseki, Shoo The transformation formula for the Dedekind modular function and related functional equations Duke Math J 24 (1957),653-662 MR 19, 943a 18 Knopp, Marvin I Modular Functions in Analytic Number Theory Markham Mathematics Series, Markham Publishing Co., Chicago, 1970 MR 42 #198 19 Lehmer, D H Ramanujan's function ,(n) Duke Math J 10 (1943), 483-492 MR 5, 35b 20 Lehmer, D H Properties of the coefficients of the modular invariant J(,) Amer J Math 64 (1942), 488-502 MR 3, 272c 21 Lehmer, D H On the Hardy-Ramanujan series for the partition function J London Math Soc 12 (1937),171-176 22 Lehmer, D H On the remainders and convergence of the series for the partition function Trans Amer Math Soc 46 (1939),362-373 MR 1, 69c 23 Lehner, Joseph Divisibility properties of the Fourier coefficients of the modular invariant}(,) Amer J Math 71 (1949), 136-148 MR 10, 357a 24 Lehner, Joseph Further congruence properties of the Fourier coefficients of the modular invariantj(')' Amer J Math 71 (1949), 373-386 MR 10, 357b 25 Lehner, Joseph, and Newman, Morris Sums involving Farey fractions Acta Arith 15 (1968/69),181-187 MR 39 # 134 26 Lehner, Joseph Lectures on Modular Forms National Bureau of Standards, Applied Mathematics Series, 61, Superintendent of Documents, U.S Government Printing Office, Washington, D.C., 1969 MR 41 #8666 27 LeVeque, William Judson Reviews in Number Theory, volumes American Math Soc., Providence, Rhode Island, 1974 28 Mordell, Louis J On Mr Ramanujan's empirical expansions of modular functions Proc Cambridge Phil Soc 19 (1917), 117-124 29 Neville, Eric H The structure of Farey series Proc London Math Soc 51 (1949), 132-144 MR 10, 681f 30 Newman, Morris Congruences for the coefficients of modular forms and for the coefficients ofj(')' Proc Amer Math Soc (1958), 609-612 MR 20 #5184 31 Petersson, Hans Ober die Entwicklungskoeffizienten der automorphen formen Acta Math 58 (1932),169-215 32 Petersson, Hans Ober eine Metrisierung der ganzen Modulformen Jber Deutsche Math 49 (1939), 49-75 33 Petersson, H Konstruktion der samtlichen Losungen einer Riemannscher Funktionalgleichung durch Dirichletreihen mit Eulersche Produktenwicklung I Math Ann 116 (1939), 401-412 Z 21, p 22; II 117 (1939),39-64 Z 22,129 34 Rademacher, Hans Ober die Erzeugenden von Kongruenzuntergruppen der Modulgruppe Abh Math Seminar Hamburg, (1929),134-148 35 Rademacher, Hans Zur Theorie der Modulfunktionen J Reine Angew Math ·167 (1932), 312-336 36 Rademacher, Hans On the partition function p(n) Proc London Math Soc (2) 43 (1937), 241-254 37 Rademacher, Hans The Fourier coefficients of the modular invariantj(r) Amer J Math 60 (1938),501-512 38 Rademacher, Hans On the expansion of th'e partition function in a series Ann of Math (2) 44 (1943), 416-422 MR 5, 35a 197 Bibliography 39 Rademacher, Hans Topics in Analytic Number Theory Die Grundlehren der mathematischen Wissenschaften, Bd 169, Springer-Verlag, New York-Heidelberg-Berlin, 1973 Z 253.10002 40 Rademacher, Hans and Grosswald, E Dedekind Sums Carus Mathematical Monograph, 16 Mathematical Association of America, 1972 Z 251 10020 41 Rademacher, Hans and Whiteman, Albert Leon Theorems on Dedekind sums Amer J Math 63 (1941),377-407 MR 2, 249f 42 Rankin, Robert A Modular Forms and Functions Cambridge University Press, Cambridge, Mass., 1977 MR 58 #16518 43 Riemann, Bernhard Gessamelte Mathematische Werke B G Teubner, Leipzig, 1892 Erliiuterungen zu den Fragmenten XXVIII Von R Dedekind, pp 466478 44 Schoeneberg, Bruno Elliptic Modular Functions Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd 203, Springer-Verlag, New York-Heidelberg-Berlin, 1974 MR 54 #236 45 Sczech, R Ein einfacher Beweis der Transformationsformel fUr log 1](z) Math Ann 237 (1978), 161-166 MR 58 #21948 46 Selberg, Atle On the estimation of coefficients of modular forms Proc Sympos Pure Math., Vol VIII, pp 1-15 Amer Math Soc., Providence, R.I., 1965 MR 32 #93 47 Serre, Jean-Pierre A Course in Arithmetic Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1973 Mathematika I (1954), 48 Siegel, Carl Ludwig A simple proofoflJ( - liT) = IJ(T) MR 16, 16b 49 Titchmarsh, E C.lntroduction to the Theory of Fourier Integrals Oxford, Clarendon Press, 1937 JTii 50 Tunin, Paul On some approximative Dirichlet polynomials in the theory of the zeta-function of Riemann DallSke Vid Selsk MaI.-Fys Medd 24 (1948), no 17, 36 pp MR 10, 286b 51 Tunin, Paul Nachtrag zu meiner Abhandlung "On some approximative Dirichlet polynomials in the theory of the zeta-function of Riemann." Acta Math A cad Sci Hungar JO (1959),277-298 MR 22 #6774 52 Uspensky, J V Asymptotic formulae for numerical functions which occur in the theory of partitions [Russian] Bull A cad Sci URSS (6) 14 (1920), 199-218 53 Watson, G N A Treatise on the Theory of Bessel Functions, 2nd Edition Cambridge University Press, Cambridge, 1962 198 Index of special symbols Q(W I , w z ) f.J(z) Gn Gz gZ,g3 e l , ez, e3 l1(Wl, w z), l1(t) H J(t) t(l1) a.(n) r S,T RG R 1/(t) s(h, k) A(X) A(IX, {3, z) '(s, a) F(x, s) j(t) ro(q) !p(t) lattice generated by WI and Wz , Weierstrass f.J-function, Eisenstein series of order 11, 11 ~ 3, Eisenstein series of order 2, invariants, values of f.J at the half-periods, discriminant g~ - 27g~, upper half-plane Im(t) > 0, Klein's modular function g~/l1, Ramanujan tau function, sum of the IXth powers of divisors of 11, modular group, generators of r, fundamental region of sub-group G of r, fundamental region of r, Dedekind eta function, Dedekind sum, -Iog(l - e- 2U ), Iseki's function, Hurwitz zeta function, periodic zeta function, 12 3J(t), congruence subgroup of r, Ip-I ('+A) IJ, P P 10 12 69 12 13 14 14 15 20 20 28 28 30 31 47 52 52 53 55 55 74 75 80 l=O ( t) 9(t) p(n) ( l1(qt)Y/(Q-I) l1(t) , Jacobi theta function, partition function, 86 91 94 199 Index of special symbols F(x) Fn Mk Mk,o T" r(11) K EZk(r) F(Z) VJ((To) 'n(s) generating function for p(Il), set of Farey fractions of order 11, linear space of entire forms of weight k, subspace of cusp forms of weight k, Hecke operator, set of transformations of order 11, dim M Zk,O, normalized Eisenstein series, Bohr function associated with Dirichlet series, set of values taken by Dirichlet series f(s) on line (T = (To, partial sums I k- S , k5n 200 94 98 117 119 120 122 133 139 168 170 185 Index A Abscissa, of absolute convergence, 165 of convergence, 165 Additive number theory, I Apostol, Tom M., 196 Approximation theorem, of Dirichlet, 143 of Kronecker, 148, 150, 154 of Liouville, 146 Asymptotic formula for p(n), 94, 104 Atkin, A O L., 91, 196 Automorphic function, 79 B Basis for sequence of exponents, 166 Bernoulli numbers, 132 Bernoulli polynomials, 54 Berwick, W E H., 22 Bessel functions, 109 Bohr, Harald, 161,196 Bohr, equivalence theorem, 178 function of a Dirichlet series, 168 matrix, 167 c Circle method, 96 Class number of quadratic form, 45 Congruence properties, of coefficients of j(r), 22, 90 of Dedekind sums, 64 Congruence subgroup, 75 Cusp form, 114 D Davenport, Harold, 136 Dedekind, Richard, 47 Dedekind function '1(r), 47 Dedekind sums, 52, 61 Deligne, Pierre, 136, 140, 196 201 Index Differential equation for p(z), II Dirichlet, Peter Gustav Lejeune, 143 Dirichlet's approximation theorem, 143 Dirichlet L-function, 184 Dirichlet series, 161 Discriminant d(-r), 14 Divisor functions a.(n), 20 Doubly periodic functions, E e l ,e ,e3 ,13 Eigenvalues of Hecke operators, 129 Eisenstein series G., 12 recursion formula for, 13 Elliptic functions, Entire modular forms, 114 Equivalence, of general Dirichlet series, 173 of ordinary Dirichlet series, 174 of pairs of periods, of points in the upper half-plane H, 30 of quadratic forms, 45 Estimates for coefficients of modular forms, 134 Euler, Leonhard, 94 Euler products of Dirichlet series, 136 Exponents of a general Dirichlet series, 161 F Farey fractions, 98 Ford, L R., 99, 196 Ford circles, 99 Fourier coefficients ofj(-r), 21, 74 divisibility properties of, 22, 74, 91 Functional equation, for "I(-r), 48, 52 for 9(-r), 91 for C(s), 140 for t\(0(, p, z), 54 for clI(lX, p, s), 56, 71 Fundamental pairs of periods, Fundamental region, of modular group 31 of subgroup r o(P), 76 r 202 G g2,g3,12 General Dirichlet series, 161 Generators, of modular group r, 28 of congruence subgroup r o(P), 78 Grosswald, Emil, 61, 198 Gupta, Hansraj, III, 196 H Half-plane H, 14 Half-plane, of absolute convergence, 165 of convergence, 165 Hardy, Godfrey Harold, 94, 196 Hardy-Ramanujan formula for pen), 94 Haselgrove, C B., 186, 196 Hecke, Erich, 114, 120, 133, 196, 197 Hecke operators T., 120 Helly, Eduard, 179 Helly selection principle, 179 Hurwitz, Adolf, 55, 145 Hurwitz approximation theorem, 145 Hurwitz zeta function, 55, 71 I InvariantsB2,B3,12 Inversion problem for Eisenstein series, 42 Iseki, Sho, 52, 197 Iseki's transformation formula, 53 J j(-r), J(-r), 74, 15 Fourier coefficients of, 21 Jacobi, Carl Gustav Jacob, 6, 91,141 Jacobi theta function, 91, 141 Jacobi triple product identity, 91 K Klein, Felix, 15 Klein modular invariant J(-r), 15 Index Kloostennan H D • 136 Knopp Marvin I., 197 Kronecker, Leopold 148 Kronecker approximation theorem, 148 150,154 L Lambert, Johann Heinrich, 24 Lambert series 24 Landau, Edmund 186 Lehmer Derrick Henry 22, 93, 95, 197 Lehmer conjecture, 22 Lehner, Joseph, 22, 91, Ill,) 97 LeVeque, William Judson, 197 Linear space Mk of entire forms, 118 Linear subspace M k of cusp forms, 119 Liouville, Joseph, 5, 146, 184 Liouville approximation theorem, 146 Liouville function ).(n), 25, 184 Liouville numbers, 147 Littlewood, John Edensor, 95 N Neville, Eric Harold, 110, 197 Newman, Morris, 91, Ill, 197 Normalized eigenform, 130 o O'Brien, J N., 91,196 Order of an elliptic function, p f.J-function of Weierstrass, 10 Partition function pen), 1, 94 Period, Period parallelogram, Periodic zeta function, 55 Petersson, Hans, 22, 133, 140, 197 Petersson inner product, 133 Petersson-Ramanujan conjecture, 140 Picard, Charles Emile, 43 Picard's theorem, 43 Product representation for A(t), 51 M Mapping properties of J(t), 40 Mediant,98 Mellin, Robert Hjalmar 54 Mellin inversion formula, 54 Mobius, Augustus Ferdinand, 24, 27, 187 Mobius function, 24, 187 Mobius transformation, 27 Modular forms, 114 and Dirichlet series, 136 Modular function, 34 Modular group r, 28 subgroups of, 46, 75 Montgomery, H L., 187 Mordell, Louis Joel, 92, 197 Multiplicative property, of coefficients of entire forms, 130 of Hecke operators, 126, 127 of Ramanujan tau function, 93, 114 Q Quadratic forms, 45 R Rademacher, Hans, 22, 62, 95, 102, 104, 197 Rademacher path of integration, 102 Rademacher series for pen), 104 Ramanujan, Srinivasa, 20, 92, 94, 136,191 Ramanujan conjecture, 136 Ramanujan tau function, 20, 22, 92, 113, 131, 198 Rankin, Robert A., 136, 198 Reciprocity law for Dedekind sums, 62 Representative of quadratic form, 45 203 Index Riemann, Georg Friedrich Bernhard, 140, 155, 185, 198 Riemann zeta function, 20, 140, 155, 18S, 189 Rouche, Eugene, 180 Rouche's theorem, 180 s Sa lie, Hans, 136 Schoeneberg, Bruno, 198 Sczech, R., 61, 198 Selberg, Atle, 136, 198 Serre, Jean-Pierre, 198 Siegel, Carl Ludwig, 48, 198 Simultaneous eigenforms, 130 Spitzenform, 114 Subgroups of the modular groups, 46, 75 T Tau function, 20, 22, 92, I 13, 131 Theta function, 91,141 Transcendental numbers, 147 Transformation of order n, 122 Transformation formula, of Dedekind, 48,52 of Iseki, 54 Tunin, Paul, 185, 198 Tunin's theorem, 185, 186 u Univalent modular function, 84 Uspensky, J V., 94, 198 V Valence of a modular function, 84 Van Wijngaarden, A., 22 Values, of J(r), 39 of Dirichlet series, 170 Vertices of fundamental region, 34 204 w Watson, G N., 109, 198 Weierstrass, Karl, Weierstrass 8c1-function, 10 Weight of a modular form, 114 Weight formula for zeros of an entire form, 115 Whiteman, Albert Leon, 62, 198 z Zeros, of an elliptic function, Zeta function, Hurwitz, 55 periodic, 55 Riemann, 140, 155, 185, 189 Zuckerman, Herbert S., 22 Graduate Texts in Mathematics (comillued from page ii) 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory FARKAS/KRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 3rd ed HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras IrTAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BURRIS/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FORSTER Lectures on Riemann Surfaces BOTT/Tu Differential Fonns in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELAND/RoSEN A Classical Introduction to Modem Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERCE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BR0NDSTED An Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups DIESTEL Sequences and Series in Banach Spaces DUBROVIN/FoMENKOlNovIKOV Modem Geometry-Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups SHIRYAEV Probability 2nd ed CONWAY A Course in Functional Analysis 2nd ed KOBLITZ Introduction to Elliptic Curves and Modular Fonns 2nd ed BROCKERIToM DIECK Representations of Compact Lie Groups GROVE/BENSON Finite Reflection Groups 2nd ed 100 BERG/CHRISTENSEN/RESSEL Hannonic Analysis on Semigroups: Theory of Positive Definite and Related Functions 101 EDWARDS Galois Theory 102 VARADARAJAN Lie Groups, Lie Algebras and Their Representations 103 LANG Complex Analysis 3rd ed 104 DUBROVIN/FoMENKOINOVIKOV Modem Geometry-Methods and Applications Part II 105 LANG SL 2(R) 106 SILVERMAN The Arithmetic of Elliptic Curves 107 OLVER Applications of Lie Groups to Differential Equations 2nd ed 108 RANGE Holomorphic Functions and Integral Representations in Several Complex Variables 109 LEHTO Univalent Functions and Teichmiiller Spaces 110 LANG Algebraic Number Theory III HUSEMOLLER Elliptic Curves 112 LANG Elliptic Functions 113 KARATZAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed 114 KOBLITZ A Course in Number Theory and Cryptography 2nd ed 115 BERGERIGOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces 116 KELLEy/SRINIVASAN Measure and Integral Vol I 117 SERRE Algebraic Groups and Class Fields 118 PEDERSEN Analysis Now 119 ROTMAN An Introduction to Algebraic Topology 120 ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAUS/HERMES et al Numbers Readings ill Mathematics 124 DUBROVIN/FoMENKOlNovIKOV Modem Geometry-Methods and Applications Part III 125 BERENSTEIN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 2nd ed 127 MASSEY A Basic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FULTON/HARRIs Representation Theory: A First Course Readings in Mathematics 130 DODSON/POSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings 132 BEARDON Iteration of Rational Functions 133 HARRIS Algebraic Geometry: A First Course 134 ROMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKINS/WEINTRAUB Algebra: An Approach via Module Theory 137 AXLERIBoURDON/RAMEY Harmonic Function Theory 2nd ed 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERIWEISPFENNING/KREDEL Grabner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNIS/FARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K- Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 EISEN BUD Commutative Algebra with a View Toward Algebraic Geometry 151 SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWN/PEARCY An Introduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAVIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEIN/ERDEL YI Polynomials and Polynomial Inequalities 162 ALPERIN/BELL Groups and Representations 163 DIXON/MORTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sum sets 166 SHARPE Differential Geometry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 2nd ed 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An Introduction to Knot Theory 176 LEE Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 CLARKEILEDYAEV/STERNlWoLENSKI Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KRESS Numerical Analysis 182 WALTER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOLLOBAS Modem Graph Theory 185 COx/LITTLElO'SHEA Using Algebraic Geometry 186 RAMAKRISHNANIVALENZA Fourier Analysis on Number Fields 187 HARRIS/MoRRISON Moduli of Curves 188 GOLDBLATT Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 ESMONDE/MuRTY Problems in Algebraic Number Theory 191 LANG Fundamentals of Differential Geometry 192 HIRSCH/LACOMBE Elements of Functional Analysis 193 COHEN Advanced Topics in Computational Number Theory 194 ENGELINAGEL One-Parameter Semi groups for Linear Evolution &}uations 195 NATHANSON Elementary Methods in Number Theory 196 OSBORNE Basic Homological Algebra 197 EISENBUD/HARRIS The Geometry of Schemes 198 ROBERT A Course inp-adic Analysis 199 HEDENMALM/KORENBLUM/ZHU Theory of Bergman Spaces 200 BAO/CHERN/SHEN An Introduction to Riemann-Finsler Geometry 201 HINDRY/SILVERMAN Diophantine Geometry: An Introduction 202 LEE Introduction to Topological Manifolds 203 SAGAN The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions 2nd ed 204 ESCOFIER Galois Theory 205 FELlXlHALPERINITHOMAS Rational Homotopy Theory 206 MURTY Problems in Analytic Number Theory Readings in Mathematics 207 GODSILlRoYLE Algebraic Graph Theory ... Congress Cataloging -in- Publication Data Apostol, Тот М Modular functions and Dirichlet series in number theory /Tom М Apostol. -2nd ed р cm.-{Graduate texts in mathematics; 41) Includes bibliographical... EDWARDS Fourier Series Vol l 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed (continued after index) Tom M Apostol Modular Functions and Dirichlet Series in Number Theory Second... GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd