Graduate Texts in Mathematics 41 Editorial Board F W Gehring P R Halmos Managing Editor c C Moore Tom M Apostol Modular Functions and Dirichlet Series in Number Theory Springer-Verlag Berlin Heidelberg GmbH 1976 Tom M Apostol Professor of Mathematics California Institute of Technology Pasadena California 91125 Editorial Board P R Halmos F W Gehring c C Moore Managing Editor University of California University of Michigan Department of Mathematics Ann Arbor, Michigan 48104 University of California at Berkeley Department of Mathematics Berkeley, California 94720 Mathematics Department Santa Barbara, California 93106 AMS Subject Classifications IOA20, l0A45, 10045, IOH05, IOHIO, IOJ20, 30AI6 Library of Congress Cataloging in Publication Data Apostol, Tom M Modular functions and Dirichlet series in number theory (Graduate texts in mathematics; 41) The second of two works evolved from a course (Mathematics 160) offered at the California Institute of Technology, continuing the subject matter ofthe author's Introduction to analytic number theory Bibliography: p 190 Includes index Numbers, Theory of Functions, Elliptic Functions, Modular Title II Series 76-10236 QA241.A62 512'.73 AII rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Veriag © 1976, Springer-Verlag Berlin Heidelberg Originally published by Springer-Verlag Inc in 1976 Softcover reprint ofthe hardcover I st edition 1976 ISBN 978-1-4684-9912-4 ISBN 978-1-4684-9910-0 (eBook) DOI 10.1007/978-1-4684-9910-0 iv Preface This is the second volume of a 2-volume textbook* which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years The second volume presupposes a background in number theory comparable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis Most of the present volume is devoted to elliptic functions and modular functions with some of their number-theoretic applications Among the major topics treated are Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular functionj( r), and Hecke's theory of entire forms with multiplicative Fourier coefficients The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series Both volumes of this work emphasize classical aspects of a subject wh ich in recent years has undergone a great deal of modern development It is hoped that these volumes will help the nonspecialist become acquainted with an important and fascinating part of mathematics and, at the same time, will provide some of the background that belongs to the repertory of every specialist in the field This volume, like the first, is dedicated to the students who have taken this course and have gone on to make notable contributions to number theory and other parts of mathematics T M A January, 1976 * The first volume is in the Springer-Verlag series Undergraduate Texts in Mathematics under the title Introduction to Analytic Number Theory v Contents Chapter I Elliptic functions 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 Introduction Doubly periodic functions Fundamental pairs of periods Elliptic functions Construction of elliptic functions The Weierstrass f.J function The Laurent expansion of f.J near the origin Differential equation satisfied by f.J The Eisenstein series and the invariants g2 and g3 The numbers e!, e2' e The discriminant ~ Klein's modular function J(r) Invariance of J under unimodular transformations The Fourier expansions of g2(r) and g3(r) The Fourier expansions of ~(r) and J( r) Exercises for Chapter I 11 11 12 13 14 15 16 18 20 23 Chapter The Modular group and modularfunctions 2.1 2.2 2.3 2.4 Möbius transformations The modular group r Fundamental regions Modular functions 26 28 30 34 Vll 2.5 2.6 2.7 2.8 2.9 Special values of J Modular functions as rational functions of J Mapping properties of J Application to the inversion problem for Eisenstein series Application to Picard's theorem Exercises Jor Chapter 39 40 40 42 43 44 Chapter The Dedekind eta function 3.1 3.2 3.3 3.4 3.5 3.6 Introduction Siegel's proof of Theorem 3.1 Infinite product representation for ß(r) The general functional equation for rt(r) Iseki's transformation formula Deduction of Dedekind's functional equation from Iseki's formula 3.7 Properties of Dedekind sums 3.8 The reciprocity law for Dedekind sums 3.9 Congruence properties of Dedekind sums 3.10 The Eisenstein series G2 (r) Exercises Jor Chapter 47 48 50 51 53 58 61 62 64 69 70 Chapter Congruences Jor the coeJJicients oJ the modular function j 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Introduction The subgroup r o(q) Fundamental region of r o(p) Functions automorphic under the subgroup r o(P) Construction of functions belonging to r o(P) The behavior of Jp under the generators of r The function q>(r) = ß(qr)jß(r) The univalent function (r) Invariance of (r) under transformations of r o(q) The functionjp expressed" as a polynomial in ExercisesJor Chapter 74 75 76 78 80 83 84 86 87 88 91 Chapter Rademacher' s series Jor the partition function 5.1 5.2 5.3 5.4 viii Introduction The plan of the proof Dedekind's functional equation expressed in terms of F Farey fractions 94 95 96 97 5.5 5.6 5.7 Ford circles Rademacher's path of integration Rademacher's convergent series for p(n) 99 102 Exercises for Chapter 110 104 Chapter Modular forms with multiplicative coefficients 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 Introduction Modular forms of weight k The weight formu1a for zeros of an entire modular form Representation of entire forms in terms of G4 and G6 The linear space M k and the subspace Mk ,o C1assification of entire forms in terms of their zeros The Hecke operators Tn Transformations of order n Behavior of Tnfunder the modular group Multiplicative property of Hecke operators Eigenfunctions of Hecke operators Properties of simu1taneous eigenforms Examp1es of norma1ized simultaneous eigenforms Remarks on existence of simultaneous eigenforms in M 2k , Estimates for the Fourier coefficients of entire forms Modular forms and Dirichlet se ries Exercises for Chapter 113 114 115 117 118 119 120 122 125 126 129 130 131 133 134 136 138 Chapter Kronecker' s theorem with applications 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Approximating real numbers by rational numbers Dirichlet's approximation theorem Liouville's approximation theorem Kronecker's approximation theorem: the one-dimensional case Extension of Kronecker's theorem to simultaneous approximation Applications to the Riemann zeta function Applications to periodic functions Exercises for Chapter 142 143 146 148 149 155 157 159 Chapter General Dirichlet series and Bohr' s equivalence theorem 8.1 8.2 8.3 Introduction The half-plane of convergence of general Dirichlet series Bases for the sequence of exponents of a Dirichlet series 161 161 166 IX 8.4 8.5 8.6 8.7 8.8 8.9 Bohr matrices The Bohr function associated with a Dirichlet se ries The set ofvalues taken by a Dirichlet seriesf(s) on a line 167 168 U 170 = Uo Equivalence of general Dirichlet series Equivalence of ordinary Dirichlet series Equality of the sets Uf(uo) and Uiuo) for equivalent Dirichlet series 8.1 The set of values taken by a Dirichlet series in a neighborhood ofthe line u = Uo 8.11 Bohr's equivalence theorem 8.12 Proof ofTheorem 8.15 8.13 Examples of equivalent Dirichlet series Applications of Bohr's theorem to L-series 8.14 Applications of Bohr's theorem to the Riemann zeta function Exercisesfor Chapter Bibliography Index of special symbols Index x 173 174 176 176 178 179 184 184 187 190 193 195 Elliptic functions 1.1 Introduction Additive number theory is concerned with expressing an integer n as a sum of integers from some given set S For example, S might consist of primes, squares, cubes, or other special numbers We ask whether or not a given number can be expressed as a sum of elements of Sand, if so, in how many ways this can be done Letf(n) denote the number of ways n can be written as a sum of elements of S We ask for various properties of f(n), such as its asymptotic behavior for large n In a later chapter we will determine the asymptotic value of the partition function p(n) which counts the number ofways n can be written as a sum of positive integers ~ n The partition function p(n) and other functions of additive number theory are intimately related to a dass of functions in complex analysis called elliptic modular functions They playa role in additive number theory analogous to that played by Dirichlet se ries in multiplicative number theory The first three chapters of this volume provide an introduction to the theory of elliptic modular functions Applications to the partition function are given in Chapter We begin with a study of doubly periodic functions 1.2 Doubly periodic functions A function f of a complex variable is called periodic with period w if f(z + w) = f(z) whenever z and z + ware in the domain off If w is aperiod, so is nw for every integer n If W and W2 are periods, so is mW l + nW for every choice of integers m and n 8: General Dirichlet series and Bohr's equivalence theorem 1(n)b(n) = for all n Therefore b(n) = with at most one exception, say b(nl)' in which case 1(nl) = O Therefore, since a(n) = b(n)e iOn , we must have a(n) = with at most one exception, say a(nl), and then 1(nl) = O Hence the series for f(s) consists of only one term,J(s) = a(n l)e - SA(nt} = a(n 1)' so f(s) itself is constant But in this ca se the theorem holds trivially D 8.l3 Examples of equivalent Dirichlet series Applications of Bohr's theorem to L-series Theorem 8.17 Let k be a given integer, and let Xbe any Dirichlet character modulo k Let L:'=l a(n)n- S be any Dirichlet series whose coefficients have the following property: a(n) #- implies (n, k) Then = I: a(~) '" I: a(n)~(n) n=l n n n=l PROOF Since these are ordinary Dirichlet series we may use Theorem 8.12 to establish the equivalence In this case we take f(n) = x(n) Then f is completely multiplicative and condition (a) is satisfied Now we show that condition (b) is satisfied We need to show that I f(p) I = if a(n) #- and pln But a(n) #- implies (n, k) = Since pln we must have (p, k) = so If (P) I = IX(p) I = since X is a character Therefore the two series are equivalent D Theorem 8.18 For a given modulus k, let 'Xl' , X (J the set of values taken by the Dirichlet L-series L(s, X;) is independent of i PROOF Applying the previous theorem with a(n) = Xl(n) we have I n= Xl~n) '" n I n= Xl(n);(n) n for every character X modulo k Here we use the fact that Xl(n) #- implies (n, k) = Thus each L-series L(s, X) is equivalent to the particular L-series L(s, Xl)' Therefore, by Bohr's theorem, L(s, X) takes the same set ofvalues as L(s, Xl) in any open half-plane within the half-plane of absolute convergence D 8.14 Applications of Bohr's theorem to the Riemann zeta function Our applications to the Riemann zeta function require the following identity involving Liouville's function 1(n) which is defined by the relations ,1(1) 184 = 1, 1(Pl a ••• p,ar) = (_I)a +···+ar 8.14: Applications ofBohr's theorem to the Riemann zeta function The function A(n) is completely multiplicative and we have (see [4], p 231) I if (J > ).(n) = ((2s) nS n=1 ((s) Theorem 8.19 Let A(n) denote Liouville's jimction and let C(x) = I ),(n) n~x n Then i[ (J > we have ((2s) = fCXc C(x) dx (s - 1)((s) XS PROOF By Abel's identity (Theorem 4.2 in [4]) we have L A(n) ~ = n,; x Keep (J > and let x C(x) C(x) n n ~ 00 x" tS + = 0(1) Then o(~ L ~) = = XS + s fX C(t) dt XS S n,;x O(log x" n x) as x~ 00, so we find ~ A(n) = s L n= nS +1 I X) S+1 C(t) dt, t for > O (J Replacing s by s - we get I n= A(7) = (s - 1) n Joo C~) dt for (J > t Since the series on the left has sum ((2s)/((s) the proof is complete D Now we prove a remarkable theorem discovered by P Tunin [44] in 1948 which gives a surprising connection between the Riemann hypothesis and the partial sums of the Riemann zeta function in the half-plane (J > Theorem 8.20 Let n (n(s) = k~1 kS • IJ there exists an no such that (n(s) "# Jor all n 2:: no and all ((s) "# Jor (J > l Ik= (J > 1, then Ik= First we note that the two Dirichlet series k- S and A(k)k- S are equivalent because A is completely multiplicative and has absolute PROOF 185 8: General Dirichlet series and Bohr's equivalence theorem value Therefore, by Bohr's theorem, (n(s) =f for A(k)k -s =f for (J > But for s real we have Ik= lim s~ + ce ±A(~) = k= k Hence for an real s > we must have we find ~ A(k) > k - L k=1 A(l) Ik= (J > implies that = 1 A(k)k -s > O Letting s + if n 2=: no· In other words, the function (11) L A(n) C(x) = n,;;x n is nonnegative for x 2=: no Now we use the identity of Theorem 8.19, (2s) (s - l)(s) = foo C(x) dx XS ' valid for (J > Note that the denominator (s - l)(s) is nonzero on the real axis s > 1, and (2s) is finite for real s > Therefore, by the integral analog of Landau's theorem (see Theorem 11.13 in [4J) the function on the left is analytic everywhere in the half-plane (J > This implies that (s) =f for (J > !, and the proof is complete D Tunin's theorem assumes that the sum C(x) in (11) is nonnegative for an x 2=: no In 1958, Haselgrove [14J proved, by an ingenious use of machine computation, that C(x) is negative for infinitely many values of x Therefore, Theorem 8.20 cannot be used to prove the Riemann hypothesis Subsequently, Turan [45J sharpened his theorem by replacing the hypothesis C(x) 2=: by a weaker inequality that cannot be disproved by machine computation Theorem 8.21 (Turan) Let C(x) = r:x > 0, C > and no such that (12) C(x» Ln,;; xA(n)/n rr there exist constants loga x -c Jx for all x 2=: no, then the Riemann hypothesis is true PROOF If t: > is given there exists an n l 2=: no such that c loga x ::;; x' for an x 2=: nl so (12) implies C(x) > - x'- 1/2 186 Exercises for Chapter Let A(x) = C(x) + X we have - f oo -sC(x) dx+ f oo x s -dx f oo -A(x) X X < dX 1 ((2s) = (s- 1)((S) + S - 21 - = f(s), say Arguing as in the proof of Theorem 8.20, we find that the function f(s) is analytic on the realline s > ! + By Landau's theorem it follows that f(s) is analytic in the half-plane (J > + This implies that ((s) =I- for (J > ! + 8, hence ((s) =I- for (J > ! since can be arbitrarily small Note Since each function (n(s) is a Dirichlet series which does not vanish identically there exists a half-plane (J > + (Jn in which (n(s) ne ver vanishes (See [4], Theorem 11.4.) The exact value of (Jn is not yet known In his 1948 paper [44] Tunin proved that, for all sufficiently large n, (is) =I- in the half-plane (J > + 2(log log n)j log n, hence (Jn ~ 2(log log n)j log n for large n In the other direction, H L Montgomery has shown that there exists a constant c > such that for all sufficiently large n, (n(s) has a zero in the halfplane (J > + c(log log n)jlog n, hence (Jn ~ c(log log n)jlog n for large n The number + (Jn is also equal to the abscissa of convergence of the Dirichlet series for the reciprocal1/(n(s) If (J > + (Jn we can write f _ (is) - where ~n(k) ~ik) k= ks' is the Dirichlet inverse of the function un(k) given by un(k) = The usual Möbius function ~(k) {I ~f k ~ n, If k > n is the limiting case of ~n(k) as n ~ 00 Exercises für Chapter If L a(n)e - s).(n) has abscissa of convergence _ I (Je - 1m sup n ~ 00 (Je < 0, prove that loglLk=n a(k) I A(n) Let (Je and (Ja denote the abscissae of convergence and absolute convergence of a Dirichlet series Prove that This gives ° S ° S (Ja - (Je (Ja - (Je log n S hm sup A(n) n~oo S for ordinary Dirichlet series 187 8: General Dirichlet series and Bohr's equivalence theorem If log n/2(n) as n -> -> 00 prove that (Ja = (Je = log Ia(n) I hm sup A (n) n~ 00 Wh at does this imply about the radius of convergence of apower series? Let {}ern)} be a sequence of camp/ex numbers Let Adenote the set of all points a(n)e-SA(n l converges absolutely Prove that A s = (J + it for which the se ries is convex I Exercises 5, 6, and refer to the seriesf(s) and coefficients given as follows Also, a(n + I,';"= = n A(n) -1 - log -1 -log a(n) 1 A(n) - log log a(n) -4 10) = Prove that (Ja = - 1 (n + log -4 -i 2- n and A(n + 10) = 2" 10 log 1 -8 2" + log -1 a(n)e-SA(n) with exponents 2" n -8 + 1) log for n ~ (log 2)/log Show that the Bohr function corresponding to the basis B = (1, log 2, log 3) is F(Zl' Z2' Z3) = cos(iztl- ti sin(iz2)(1 + cos(iztl) + - 2e- z , , - e z, if X3 > -log 2, z 1, Z2 arbitrary Determine the set V f(O) H int: The points - 1, + i, - i are significant Assume the Dirichlet series f(s) = H (J > (Ja prove that lim ~ T~hJ2T fT -T Assume the series f(s) v(n) = eA(n+it1f((J = I.':o= a(n)e-SA(n) converges absolutely for (J + it) dt = if A = A(n) ifA=lA(1),A(2), I;;"=l a(n)e-sA(nl converges absolutely for (J > (Ja > O Let eA(n l (a) Prove that the series g(s) = (b) If (J > (Ja prove that I::,= a(n)e-sv(nl converges absolutely if (J > r(s)f(s) = 188 farn) 00 > (Ja g(t)t,-l dt O Exercises for Chapter This extends the c1assic formula for the Riemann zeta function, f(s)((s) Hint: First show that f(s)e-SA(n) = = f02 ~ dt a e - So e-tv(n)t s- dt 189 Bibliography Apostol, Tom M Sets ofvalues taken by Dirichlet's L-series Proe Sympos Pure Math., Vol VIII, 133-137 Amer Math Soc., Providence, R.I., 1965 MR 31 # 1229 Apostol, Tom M Caleulus, Vol 11, 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 Analytie Number Theory Undergraduate Texts in Mathematics Springer-Verlag, New York, 1976 Atkin, A O L and O'Brien, J N Some properties ofp(n) and e(n) modulo powers of 13 Trans Amer Math Soe 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 Weil I Inst haut Etud sei., Publ math 43 (1973), 273-307 (1974) Z 287,14001 Erdös, P A note on Farey series Quart J Math., Oxford Sero 14 (1943), 82-85 MR 5, 236b Ford, Lester R Fractions Amer Math Monthly 45 (1938),586-601 10 Gantmacher, F R The Theory of Matriees, Vo! Chelsea Pub! Co., New York, 1959 11 Gunning, R C Leetures 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 formu1ae in combinatory analysis Proe London Math Soe (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 Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung Math Ann 112 (1936),664-699 190 Bibliography 16 Hecke, E Über Modulfunktionen und die Dirichlet Reihen mit Eulerscher Produktentwicklung I Math Ann.1l4 (1937), 1-28; Il 316-351 17 Iseki, Sho The transformation formula for the Dedekind modular function and related functional equations Duke Math J 24 (1957), 653-662 MR 19, 943a 18 Lehmer, D H Ramanujan's function r(n) Duke Math J 10 (1943), 483-492 MR 5, 35b 19 Lehmer, D H Properties of the coefficients of the modular invariant J(r) Amer J Math 64 (1942), 488-502 MR 3, 272c 20 Lehmer, D H On the Hardy-Ramanujan se ries for the partition function J London Math Soc 12 (1937),171-176 21 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 22 Lehner, Joseph Divisibility properties of the Fourier coefficients of the modular invariant}(r) Amer J Math 71 (1949),136-148 MR 10, 357a 23 Lehner, Joseph Further congruence properties of the Fourier coefficients of the modular invariant}(r) Amer J Math 71 (1949), 373-386 MR 10, 357b 24 Lehner, Joseph, and Newman, Morris Sums involving Farey fractions Acta Arith 15 (1968/69), 181-187 MR 39 # 134 25 Lehner, Joseph Lectures on Modular Forms National Bureau of Standards, Applied Mathematics Series, 61, Superintendent ofDocuments, V.S Government Printing Office, Washington, D.C., 1969 MR 41 #8666 26 LeVeque, William Judson Reviews in Number Theory, volumes American Math Soc., Providence, Rhode Island, 1974 27 MordelI, Louis J On Mr Ramanujan's empirical expansions ofmodular functions Proc CambridgePhil Soc.19(1917), 117-124 28 Neville, Eric H The structure of Farey series Proc London Math Soc 51 (1949), 132-144 MR 10, 681f 29 Newman, Morris Congruences for the coefficients of modular forms and for the coefficients of}(r) Proc Amer Math Soc (1958),609-612 MR 20 #5184 30 Petersson, Hans Über die Entwicklungskoeffizienten der automorphen formen Acta Math 58 (1932), 169-215 31 Petersson, Hans Über eine Metrisierung der ganzen Modulformen Jber Deutsche Math 49 (1939),49-75 32 Petersson, H Konstruktion der sämtlichen Lösungen einer Riemannscher Funktionalgleichung durch Dirichletreihen mit Eulersche Produktenwicklung I Math Ann 116 (1939), 401-412 Z 21, p 22; 11.117 (1939),39-64 Z 22,129 33 Rademacher, Hans Über die Erzeugenden von Kongruenzuntergruppen der Modulgruppe Abh Math Seminar Hamburg, (1929), 134-148 34 Rademacher, Hans On the partition function p(n) Proc London Math Soc (2) 43 (1937),241-254 35 Rademacher, Hans The Fourier coefficients of the modular invariant}(r) Amer J Math 60 (1938),501-512 36 Rademacher, Hans On the expansion of the partition function in aseries Ann Math (2) 44 (1943),416-422 MR 5, 35a 0/ 37 Rademacher, Hans Topics in Analytic Number Theory Die Grundlehre der mathematischen Wissenschaften, Bd 169, Springer-Verlag, New York-HeidelbergBerlin, 1973 Z 253.10002 191 Bibliography 38 Rademacher, Hans and Grosswald, E Dedekind Sums Carus Mathematical Monograph, 16 Mathematical Association of America, 1972 Z 251 10020 39 Rademacher, Hans and Whiteman, Albert Leon Theorems on Dedekind sums Amer J Math 63 (1941),377-407 MR 2, 249f 40 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 41 Serre, Jean-Pierre A Course in Arithmetic Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1973 42 Siegel, Carl Ludwig A simple proof off/( -I/T) = fleT) JT{i Mathematika (1954), MR 16, 16b 43 Titchmarsh, E C.Introduction to the Theory 0/Fourier Integrals Oxford, Clarendon Press, 1937 44 Tunin, Paul On some approximative Dirichlet polynomials in the theory of the zeta-function of Riemann Danske Vid Selsk Mat.-Fys Medd 24 (1948), no 17, 36 pp MR 10, 286b 45 Tunin, Paul Nachtrag zu meiner Abhandlung "On so me approximative Dirichlet polynomials in the theory of the zeta-function of Riemann." Acta Math Acad Sei Hungar 10 (1959), 277-298 MR 22 #6774 46 Uspensky, J V Asymptotic formulae for numerical functions which occur in the theory of partitions [Russian] Bu!! Acad Sei URSS (6) 14 (1920), 199-218 47 Watson, G N A Treatise on the Theory 0/Bessel Functions, 2nd Edition Cambridge University Press, Cambridge, 1962 192 Index of special symbols Q(W , W2) IJ(z) Go G2 92,93 e , e2, e3 d(Wb w2 ), d(T) H J(T) T(n) (T.(n) r lattice generated by Wl and W , Weierstrass IJ-function, Eisenstein series of order n, n ~ 3, Eisenstein series of order 2, invariants, values of IJ at the half-periods, discriminant 9~ - 279~ , upper half-plane Im(T) > 0, Klein's modular function 9Yd, Ramanujan tau function, sum of the ath powers of divisors of n, modular group, generators of r, fundamental region of sub-group G of r, fundamental region of r, Dedekind eta function, Dedekind sum, -log(1 - e- 2nX ), Iseki's function, Hurwitz zeta function, periodic zeta function, 12 J(T), congruence subgroup of r, 10 12 69 12 13 14 14 15 20 20 28 28 30 31 47 52 52 53 55 55 74 75 !p(T) tp-l (T+A) I'!, P P 80 S, T RG R I1(T) s(h, k) A(x) A(a, ß, z) ((s, a) F(x, s) j(T) ro(q) A=O ( T) 9(T) p(n) ( d(qT)y/(q-l) d(T) , Jacobi theta function, partition function, 86 91 94 193 Index of special symbols F(x) Fn Mk Mk,o T" r(n) K E 2ir:) F(Z) Vf(O'o) 'n(s) generating function for p(n), set of Farey fractions of order n, linear space of entire forms of weight k, subspace of cusp forms of weight k, Hecke operator, set of transformations of order n, dimM 2k ,0, normalized Eisenstein series, Bohr function associated with Dirichlet series, set ofvalues taken by Dirichlet series f(s) on line partial sums I k- S , k~" 194 0' = 0'0' 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., 190 Approximation theorem, ofDirichlet, 143 of Kronecker, 148, 150, 154 of Liouville, 146 Asymptotic formula for p(n), 94, 104 Atkin, A O L., 91, 190 Automorphic function, 79 B Basis for sequence of exponents, 166 Bernoulli numbers, 132 Bernoullil'olynomials, 54 Berwick, W E H., 22 Bessel functions, 109 Bohr, Harald, 161, 190 Bohr, equivalence theorem, 178 function of a Dirichlet series, 168 matrix, 167 c Circle method, 96 C1ass number of quadratic form, 45 Congruence properties, of coefficients of }(r), 22, 90 of Dedekind sums, 64 Congruence subgroup, 75 Cusp form, 114 o Davenport, Harold, 136 Dedekind, Richard, 47 Dedekind function rJ(r), 47 Dedekind sums, 52, 61 Deligne, Pierre, 136, 140, 190 195 Index Differential equation for $o(z), II Dirichlet, Peter Gustav Lejeune, 143 Dirichlet's approximation theorem, 143 Dirichlet L-function, 184 Dirichlet series, 161 Discriminant ~(r), 14 Divisor functions 0"7(n), 20 Doubly periodic functions, E et e2,e3,13 Eigenvalues of Hecke operators, 129 Eisenstein series Gn , 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 se ries, 136 Exponents of a general Dirichlet series, 161 F Farey fractions, 98 Ford, L R , 99, 190 F ord circJes, 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 ((s), 140 for I\(IX, ß, z), 54 for (IX, ß, s), 56, 71 Fundamental pairs of periods, Fundamental region, of modular group r,31 of subgroup r o(P), 76 196 G g2,g3,12 General Dirichlet series, 161 Generators, ofmodular group r, 28 of congruence subgroup r o(P), 78 Grosswald, Emil, 61,192 Gupta, Hansraj, 111, 190 H Half-plane H, 14 Half-plane, of absolute convergence, 165 of convergence, 165 Hardy, Godfrey Harold, 94, 190 Hardy-Ramanujan formula for p(n), 94 Haselgrove, C B., 186, 190 Hecke, Erich, 114, 120, 133, 190, 191 Hecke operators Tn , 120 Helly, Eduard, 179 Helly selection principle, 179 Hurwitz, Adolf, 55, 145 Hurwitz approximation theorem, 145 Hurwitz zeta function, 55, 71 I Invariants 92,93' 12 Inversion problem for Eisenstein series,42 Iseki, Shö, 52, 191 Iseki's transformation formula, 53 J j(r), J(r), 74,15 Fourier coefficients of, 21 Jacobi, earl Gustav Jacob, 6, 91,141 Jacobi theta function, 91,141 Jacobi tri pIe product identity, 91 K Klein, Felix, 15 Klein modular invariant J( r), 15 Index Kloosterman, H D., 136 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, 95,191 Lehmer conjecture, 22 Lehner, Joseph, 22, 91, Ill, 191 LeVeque, William Judson, 191 Linear space M k of entire forms, 118 Linear subspace M k • O of cusp forms, 119 Liouville, Joseph, 5, 146, 184 Liouville approximation theorem, 146 Liouville function ).(n), 25, 184 Liouville numbers, 147 Littiewood, John Edensor, 95 N Neville, Eric Harold, 110, 191 Newman, Morris, 91, Ill, 191 Normalized eigen form, 130 o O'Brien, J N., 91,190 Order of an elliptic function, p f.J-function of Weierstrass, 10 Partition functionp(n), 1,94 Period, I Period parallelogram, Periodic zeta function, 55 Petersson, Hans, 22, 133, 140, 191 Petersson inner product, 133 Petersson-Ramanujan conjecture, 140 Picard, Charles Emile, 43 Picard's theorem, 43 Product representation for ,1(T), 51 M Mapping properties of J(T), 40 Mediant, 98 MelIin, Robert Hjalmar, 54 Mellin inversion formula, 54 Möbius, Augustus Ferdinand, 24, 27, 187 Möbius function, 24, 187 Möbius transformation, 27 Modular farms, 114 and Dirichlet series, 136 Modular function, 34 Modular group r, 28 subgroups of, 46, 75 Montgomery, H L., 187 MordelI, Louis JoeJ, 92, 191 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, 191 Rademacher path of integration, 102 Rademacher series for p(n), 104 Ramanujan, Srinivasa, 20, 92, 94, 136, 191 Ramanujan conjecture, 136 Ramanujan tau function, 20, 22, 92, 113,131 Rankin, Robert A., 136 Reciprocity law for Dedekind sums, 62 Representative of quadratic farm, 45 197 Index Riemann, Georg Friedrich Bernhard, 140, 155, 185 Riemann zeta function, 20, 140, 155, 185,189 Rouche, Eugene, 180 Rouche's theorem, 180 s Salie, Hans, 136 Selberg, Atle, 136, 192 Serre, Jean-Pierre, 192 Siegel, earl Ludwig, 48, 192 Simultaneous eigenforms, 130 Spitzen form, 114 Subgroups of the modular groups, 46, 75 T Tau function, 20, 22, 92, 113, 131 Theta function, 91, 141 Transcendental numbers, 147 Transformation of order n, 122 Transformation formu1a, of Dedekind, 48, 52 ofIseki,54 Tun1n, Paul, 185, 192 Tunin's theorem, 185, 186 u Univalent modular function, 84 Uspensky,1 V., 94, 192 V Valence of a modular function, 84 Van Wijngaarden, A., 22 Values, of J(r), 39 of Dirichlet series, 170 Vertices offundamental region, 34 198 w Watson, G N , 109, 192 Weierstrass, Karl, Weierstrass p-function, 10 Weight of a modular form, 114 Weight formula for zeros of an entire form, 115 Whiteman, Albert Leon, 62, 192 z Zeros, of an elliptic function, Zeta function, Hurwitz, 55 periodic, 55 Riemann, 140, 155, 185, 189 Zuckerman, Herbert S., 22 ... Texts in Mathematics under the title Introduction to Analytic Number Theory v Contents Chapter I Elliptic functions 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 1. 10 1. 11 1 .12 1. 13 1. 14 1. 15 Introduction... simultaneous eigenforms in M 2k , Estimates for the Fourier coefficients of entire forms Modular forms and Dirichlet se ries Exercises for Chapter 11 3 11 4 11 5 11 7 11 8 11 9 12 0 12 2 12 5 12 6 12 9 13 0... Texts in Mathematics 41 Editorial Board F W Gehring P R Halmos Managing Editor c C Moore Tom M Apostol Modular Functions and Dirichlet Series in Number Theory Springer-Verlag Berlin Heidelberg GmbH