Complex analysis e freitag r busam ( 2005) WW

Eberhard Freitag Rolf Busam Complex Analysis ABC Professor Dr Eberhard Freitag Dr Rolf Busam Translator Dr Dan Fulea Faculty of Mathematics Institute of Mathematics University of Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: freitag@mathi.uni-heidelberg.de busam@mathi.uni-heidelberg.de Faculty of Mathematics Institute of Mathematics University of Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: dan@mathi.uni-heidelberg.de Mathematics Subject Classification (2000): 30-01, 11-01, 11F11, 11F66, 11M45, 11N05, 30B50, 33E05 Library of Congress Control Number: 2005930226 ISBN-10 3-540-25724-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25724-0 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11396024 40/TechBooks 543210 In Memoriam Hans Maaß (1911–1992) Preface to the English Edition This book is a translation of the forthcoming fourth edition of our German book “Funktionentheorie I” (Springer 2005) The translation and the LATEX files have been produced by Dan Fulea He also made a lot of suggestions for improvement which influenced the English version of the book It is a pleasure for us to express to him our thanks We also want to thank our colleagues Diarmuid Crowley, Winfried Kohnen and Jă org Sixt for useful suggestions concerning the translation Over the years, a great number of students, friends, and colleagues have contributed many suggestions and have helped to detect errors and to clear the text The many new applications and exercises were completed in the last decade to also allow a partial parallel approach using computer algebra systems and graphic tools, which may have a fruitful, powerful impact especially in complex analysis Last but not least, we are indebted to Clemens Heine (Springer, Heidelberg), who revived our translation project initially started by Springer, New York, and brought it to its final stage Heidelberg, Easter 2005 Eberhard Freitag Rolf Busam Contents I Differential Calculus in the Complex Plane C I.1 Complex Numbers I.2 Convergent Sequences and Series I.3 Continuity I.4 Complex Derivatives I.5 The Cauchy–Riemann Differential Equations 9 24 36 42 48 II Integral Calculus in the Complex Plane C II.1 Complex Line Integrals II.2 The Cauchy Integral Theorem II.3 The Cauchy Integral Formulas 71 72 79 94 III Sequences and Series of Analytic Functions, the Residue Theorem 105 III.1 Uniform Approximation 106 III.2 Power Series 111 III.3 Mapping Properties for Analytic Functions 126 III.4 Singularities of Analytic Functions 136 III.5 Laurent Decomposition 145 A Appendix to III.4 and III.5 158 III.6 The Residue Theorem 165 III.7 Applications of the Residue Theorem 174 IV Construction of Analytic Functions 195 IV.1 The Gamma Function 196 IV.2 The Weierstrass Product Formula 214 IV.3 The Mittag–Leffler Partial Fraction Decomposition 223 IV.4 The Riemann Mapping Theorem 228 A Appendix : The Homotopical Version of the Cauchy Integral Theorem 239 B Appendix : The Homological Version of the Cauchy Integral Theorem 244 X Contents C Appendix : Characterizations of Elementary Domains 249 V Elliptic Functions 257 V.1 The Liouville Theorems 258 A Appendix to the Definition of the Periods Lattice 265 V.2 The Weierstrass ℘-function 267 V.3 The Field of Elliptic Functions 274 A Appendix to Sect V.3 : The Torus as an Algebraic Curve 279 V.4 The Addition Theorem 287 V.5 Elliptic Integrals 292 V.6 Abel’s Theorem 299 V.7 The Elliptic Modular Group 310 V.8 The Modular Function j 319 VI Elliptic Modular Forms 327 VI.1 The Modular Group and Its Fundamental Region 328 VI.2 The k/12-formula and the Injectivity of the j-function 335 VI.3 The Algebra of Modular Forms 345 VI.4 Modular Forms and Theta Series 348 VI.5 Modular Forms for Congruence Groups 362 A Appendix to VI.5 : The Theta Group 374 VI.6 A Ring of Theta Functions 381 VII Analytic Number Theory 391 VII.1 Sums of Four and Eight Squares 392 VII.2 Dirichlet Series 409 VII.3 Dirichlet Series with Functional Equations 418 VII.4 The Riemann ζ-function and Prime Numbers 431 VII.5 The Analytic Continuation of the ζ-function 439 VII.6 A Tauberian Theorem 446 VIII Solutions to the Exercises 459 VIII.1 Solutions to the Exercises of Chapter I 459 VIII.2 Solutions to the Exercises of Chapter II 471 VIII.3 Solutions to the Exercises of Chapter III 476 VIII.4 Solutions to the Exercises of Chapter IV 488 VIII.5 Solutions to the Exercises of Chapter V 496 VIII.6 Solutions to the Exercises of Chapter VI 505 VIII.7 Solutions to the Exercises of Chapter VII 513 References 523 Symbolic Notations 533 Index 535 Introduction The complex numbers have their historical origin in the 16th century when they were created during attempts to solve algebraic equations G Cardano √ (1545) has already introduced formal expressions as for instance ± −15, in order to express solutions of quadratic and cubic equations Around 1560 R Bombelli computed systematically using such expressions and found as a solution of the equation x3 = 15x + in the disguised form 4= 2+ √ −121 + 2− √ −121 Also in the work of G.W Leibniz (1675) one can find equations of this kind, e.g √ √ √ + −3 + − −3 = √ In the year 1777 L Euler introduced the notation i = −1 for the imaginary unit The terminology “complex number” is due to C.F Gauss (1831) The rigorous introduction of complex numbers as pairs of real numbers goes back to W.R Hamilton (1837) Sometimes it is already advantageous to introduce and make use of complex numbers in real analysis One should for example think of the integration of rational functions, which is based on the partial fraction decomposition, und therefore on the Fundamental Theorem of Algebra: Over the field of complex numbers any polynomial decomposes as a product of linear factors Another example for the fruitful use of complex numbers is related to Fourier series Following Euler (1748) one can combine the real angular functions sine and cosine, and obtain the “exponential function” eix := cos x + i sin x Then the addition theorems for sine and cosine reduce to the simple formula Introduction ei(x+y) = eix eiy In particular, eix n = einx holds for all integers n The Fourier series of a sufficiently smooth function f , defined on the real line, with period 1, can be written in terms of such expressions as ∞ an e2πinx f (x) = n=−∞ Here it is irrelevant whether f is real or complex valued In these examples the complex numbers serve as useful, but ultimatively dispensable tools New aspects come into play when we consider complex valued functions depending on a complex variable, that is when we start to study functions f : D → C with two-dimensional domains D systematically The dimension two is ensured when we restrict to open domains of definition D ⊂ C Analogously to the situation in real analysis one introduces the notion of complex differentiability by requiring the existence of the limit lim f (a) := z→a z=a f (z) − f (a) z−a for all a ∈ D It turns out that this notion behaves much more drastically then real differentiability We will show for instance that a (first order) complex differentiable function is automatically arbitrarily often complex differentiable We will see more, namely that complex differentiable functions can always be developed locally as power series For this reason, complex differentiable functions (defined on open domains) are also called analytic functions “Complex analysis” is the theory of such analytic functions Many classical functions from real analysis can be analytically extended to complex analysis It turns out that these extensions are unique, as for instance in the case ex+iy := ex eiy From the relation e2πi = it follows that the complex exponential function is periodic with the purely imaginary period 2πi This observation is fundamental for the complex analysis As a consequence one can observe further phenomena: The complex logarithm cannot be introduced as the unique inverse function of the exponential function in a natural way It is a priori determined only up to a multiple of 2πi Introduction The function 1/z (z = 0) does not have any primitive in the punctured complex plane A related fact is the following: the path integral of 1/z with respect to a circle line centered in the origin and oriented anticlockwise yields the non-zero value |z|=r dz = 2πi z (r > 0) Central results of complex analysis, like e.g the Residue Theorem, are nothing but a highly generalized version of these statements Real functions often show their true nature first after considering their analytic extensions For instance, in the real theory it is not directly transparent why the power series representation = − x2 + x4 − x6 ± · · · + x2 is valid only for |x| < In the complex theory this phenomenon becomes more understandable, simply because the considered function has singularities in ±i Then its power series representation is valid in the biggest open disk excluding the singularities, namely the unit disk In the real theory it is also hard to understand why the Taylor series around of the C ∞ function e−1/x , 0, f (x) = x=0, x=0, converges for all x ∈ R, but does not represent the function in any point other than zero In the complex theory this phenomenon becomes understandable, because the function e−1/z has an essential singularity in zero Less trivial examples are more impressive Here, one should mention the Riemann ζ-function ∞ ζ(s) = n−s , n=1 which will be extensively studied in the last chapter of the book as a function of the complex variable traditionally denoted by s using the methods of complex analysis, which will be presented throughout the preceeding chapters From the analytical properties of the ζ-function we will deduce the Prime Number Theorem Riemann’s celebrated work on the ζ function [Ri2] is a brilliant example for the thesis he already presented eight years in advance in his dissertation [Ri1] VIII.6 Solutions to the Exercises of Chapter VI 505 We have lim Im τ →∞ ∆(τ )/q = a1 Because ∆(τ ) and q are real on the imaginary axis, the value of a1 is real If a1 is positive, then lim j(iy) = +∞, y→∞ lim j(iy + 1/2) = −∞ y→∞ The claim follows from the Intermediate Value Theorem for continuous functions In case of a1 < we can proceed analogously (In fact, the latter case does not occur.) Make use of Exercise from V.7 The same proof transposes word by word if we replace Γ = SL(2, Z) by the subgroup generated by the two specified matrices VIII.6 Solutions to the Exercises of Chapter VI Exercises in Sect VI.1 We have M i = i ⇐⇒ (ai + b) = i(ci + d) ⇐⇒ a = d, b = −c Because of the formula M −1 = d −b −c a the above property [ a = d , b = −c ] is equivalent to M = M −1 (a) We can suppose w = i, and use the formula √ i= y −1 √ y −x z (b) The given map is well-defined, the injectivity follows from Exercise 1, and the surjectivity from 2(a) The map is continuous by the definition of the quotient topology To obtain that it is a homeomorphism, we show that it is open, which is enough The canonical group action is transitive, so it is enough to show that the image of a neighborhood U of E ∈ SL(2, R) by the map M → M i is a neighborhood of i ∈ H For this, we can even restrict to upper triangular matrices (c = 0) in U One direction is trivial, namely, if M is elliptic, then M admits a fixed point, which is of course also a fixed point of any power of M The converse is slightly more difficult First we observe that the eigenvalues of elliptic matrices have always the modulus This is so, because the characteristic polynomial (a − λ)(d − λ) − bc = has a special shape has the free coefficient ad − bc = 1, and |a + d| < Let now M l be elliptic, and different from ±E We consider an eigenvalue ζ of M 506 VIII Solutions to the Exercises Then ζ l is an eigenvalue of M l It has modulus 1, and thus ζ has also modulus Since the eigenvalues of the real matrix M are building a pair of complex conjugated numbers, the numbers ζ and ζ = ζ −1 are the two eigenvalues of M Because of l |σ(M l )| = |ζ l + ζ | < , ζ is not ±1 This implies |σ(M )| = |ζ + ζ| < , and M is elliptic We can assume that i is the fixed point The claim translates as follows Any finite subgroup of SO(2, R) is cyclic But this group is isomorphic to the group S of all complex numbers of modulus 1, an isomorphism is given by the law eiφ −→ cos ϕ sin ϕ − sin ϕ cos ϕ The group S is isomorphic to the additive group R/Z If G ⊂ S is a finite subgroup, then its preimage in R (with respect to the projection R R/Z) is a discrete subgroup The claim now follows from the fact that any discrete subgroup of R is cyclic Exercises in Sect VI.2 From f (z) = f (M z) we get by the chain rule f (z) = f (M z)M (z) = f (M z)(cz + d)−2 g f An analytic function f : D → C is injective in a suitable neighborhood of a point a ∈ D, iff its derivative in a does not vanish, III.3 The j-function is injective modulo SL(2, Z) It is thus injective in a small neighborhood of a given point a ∈ H, iff this point is not a fixed point of the elliptic modular group We have f g − g f = −f A bijective map between topological spaces is a homeomorphism, iff it is continuous and open By the definition of the quotient topology, the j-function implements a continuous map from H/Γ to C This map is open by the Open Mapping Theorem We call two point of the fundamental region equivalent, iff there exists a modular substitution mapping one point into the other (and conversely) This gives an equivalence relation ∼ on F First it is clear that the quotient topological space F / ∼ of F with respect to this relation and H/Γ are topologically equivalent We already know the possibilities for (non-equal) equivalent points in the fundamental region F , so we can map F onto a full square without one vertex, and the transported relation ∼ identifies corresponding boundary points, which are symmetric with respect to the diagonal through the missing vertex From the topology it is known that the corresponding identification on the full square VIII.6 Solutions to the Exercises of Chapter VI 507 give rise to a sphere, removing one point we obtain a sphere without a point, i.e homeomorphically the real plane Despite of intuitive evidence, it is not easy to convert these arguments to a complete rigorous proof This Exercise may seem unfair from this point of view In the next book we will develop, in connection with the topological classification of surfaces, instruments to attack such questions The most simple argument uses the j-function Because of j(z) = j(−z) the quotient space is topologically equivalent to the quotient space of C with respect to the relation that identifies pairs of points w and w The upper and the lower half-planes are thus folded together, and the result is the closed upper halfplane (In contrast with Exercise 5, it is easy to prove it.) Exercises in Sect VI.3 Induction on the weight To start the induction process, we can take the weight k = The inductive step uses the fact, that any modular form of positive weight without zeros in the upper half-plane is necessary a cusp form, so we can divide by ∆ We study the linear map [Γ, k] −→ Cdk , which associates to a modular form the tuple of the first dk coefficients in its power series representation The Exercise claims that this map is bijective Since both vector spaces have the same dimension, it is enough to show it is injective Let f be in the kernel Then f /∆dk is an entire modular form of weight k − 12dk From the formula for dk we inductively deduce, that in this weight there exists no entire non-zero modular form Since j takes infinitely many values, P has infinitely many zeros The same applies (with the same argument, or as a consequence) for the function G34 /G26 β Let 4α+6β=k Cαβ Gα G6 be a non-trivial linear relation By multiplying with a suitable monomial we can assume that k is divisible by We divide this k/6 relation by G6 , and obtain a linear relation between powers of G34 /G26 The Ansatz G34 − C∆ works Since a meromorphic modular form f has only finitely many poles in the fundamental region, there exists by Exercise an entire modular form h, such that f h has no poles in the fundamental region, and hence in the whole H After multiplying with a suitable power of the discriminant, we obtain that g = hf is also regular in i∞ We can represent a given modular function f as in Exercise as a quotient f = g/h of two entire modular forms g, h of the same weight We can furthermore arrange that the common weight of g and h is divisible by 6, let’s say it is 6k Because of the formula g/h = (g/Gk6 )(h/Gk6 )−1 we can suppose h = Gk6 Expressing g as a polynomial in G4 and G6 we see that any modular function can be written as a rational function in G34 /G26 , and hence in j Exercises in Sect VI.4 Apply the transformation behavior for the imaginary part, V.7.1 508 VIII Solutions to the Exercises If f is a cusp form, then exp(−2πiz)f (z) is bounded in domains of the shape y ≥ δ > From the integral representation an = f (z)e−2πinz dx we deduce concretely |an | ≤ C e2π nk/2 A simple estimation shows that the R.H.S in the asserted asymptotic formula is greater than δnm/2−1 for a suitable δ > The difference of the two sides has by Exercise a smaller (dominated) asymptotic, namely O nm/4 Because of m ≥ we have m/4 < m/2 − If we subtract from a modular form a suitable constant multiple of the Eisenstein series, then we obtain a cusp form Using Exercise 3, we only have to prove the claim for the Eisenstein series This follows easily from the formula mentioned in Exercise The rows of orthogonal matrices have the Euclidian length If the matrix has integer entries, the rows are up to sign the canonical basis vectors We can list all integer orthogonal matrices U , by writing all n canonical basis vectors (as row vectors) in arbitrary order as matrix rows (there are n! possibilities for this), and for each such permutation matrix we can modify the 1-entries up to sign (in each case 2n possible modifications) There are totally 2n n! possibilities Make use of the following fact If A is an integer n × n matrix with non-zero determinant, then L = AZn is a sublattice of Zn of index |det A| The square of this number is the Gram determinant of the associated quadratic form (a) We consider first Ln ∩ Zn This is the kernel of the homomorphism Zn −→ Z/2Z , x −→ x1 + · · · + xn mod , and thus a sublattice of index of Zn For odd n we have Ln ⊂ Zn , the determinant of a Gram matrix is thus 22 = if n is even, then the vector e = (1/2, −1/2, , 1/2, −1/2) lies in Ln , and any vector a of Ln is of the shape a = b or a = e + b with b ∈ Ln ∩ Zn The index of Ln ∩ Zn in Ln is thus 2, and the determinant of a Gram matrix is hence The lattice Ln is thus of type II, iff [ n is even and a, a is even for all a ∈ Ln ] Setting a = e + b, we have a, a ≡ n/4 mod The lattice Ln is thus of type II, iff n is divisible by (b) There are two types of minimal vectors: the integer minimal vectors have twice the entry ±1, and zeros else, the non-integer minimal vectors exist only in case n = They contain only ±1/2 entries (c) The following argument is simpler as the hint: L8 , and thus also L8 × L8 , is generated by minimal vectors, but this is false for L16 Express ϑa,b in terms of the Jacobi theta function (V.6), ϑa,b (z) = eπia ϑ(z, b + za) , VIII.6 Solutions to the Exercises of Chapter VI 509 and exploit the knowledge of the zeros (Exercise in Sect V.6) If the series vanishes, then we must have b + za = β α + z, 2 α≡β≡1 mod Exactly this case was excluded If we change b modulo 1, then the value of the theta series does not change The substitution a → a + α, α ∈ Z, translates in terms of the summation index as n → n − α The series inherits thus the factor exp(−2πiab) As in Exercise 8, express the theta zero value in terms of the Jacobi theta series, and apply the Jacobi theta transformation formula 10 Let M be the mentioned finite set of pairs (a, b) We first show, that for any modular substitution M there exists a bijective self-map (a, b) → (α, β) of M with the property √ ϑa,b (M z) = v(M, a, b) cz + dϑα,β (z) , which involves a suitable unit root of order eight v(M, a, b) It is enough to show this only for the generators of the modular group For the involution we obtain this property from (both parts of) Exercise Together with (a, b) ∈ M there is also a suitable translate of (b, −a) an element in M For the translation the claim is elementary A suitable power of ∆n (z) is a modular form without zeros, and hence by Exercise in Sect VI.3 it is a constant multiple of a 4n2 −1 , and hence f discriminant power Because of this a power of f = ∆24 n /∆ itself, is constant, ∆n (z)24 = C∆(z)4n −1 To determine the constant C we develop both sides as power series in exp πiz 4n and compare the lower order coefficients Here, we use the fact that the coefficient of lowest order of a product of two power series is the product of the lowest coefficients of the involved power series We obtain C · (2π)12(4n −1) + e−πib/n = 24 = (2n)24 0≤b 0, using the convergence of an we can find an N , such that |A(n, m)| ≤ ε for m > n ≥ N Using the initial estimation we then obtain |S(n, m)| ≤ ε + |s| σ m−1 ν −σ − (ν + 1)−σ ν=n νµ+1 p µ 1 z + f p p p p−1 f ν=1 z+ν p Up to some necessary prefactors, the involution switches over the first two terms, and the terms of the big sum are permuted by Exercise For the proof, make use of the formula from Exercise and of p One checks the transformation behavior with respect to the generators For the translation z → z + the function f (pz) remains unchanged, and the terms of the sum are permuted To see the effect of the involution, we write better (T (p)f (z) = pk−1 f (pz) + |s| −σ n − m−σ σ From this we deduce the uniform convergence in regions, where |s| /σ is bounded from above Supplement The inequality σ0 ≥ σ1 is trivial If the series converges in a point s, then the sequence (an n−s ) is bounded Then we have that the Dirichlet series converges absolutely in s + + ε for any positive ε From this we deduce the second inequality We must prove these relations for the divisors power sums a(n) = σk−1 (n) The relation in (a) follows from that the fact, that any divisor of mn can be uniquely written as a product of a divisor of m and a divisor of n (m, n are relatively prime) For the relation (b) us the fact, that the divisors of pν are exactly the p-powers pj , j ≤ ν −1 −1 and 1−pk−1−s as geometric series, and multiplying Developing 1−p−s ∞ them, we obtain a series the shape ν=0 b(pν )p−νs A direct computation gives b(pν ) = σk−1 (pν ) The remained part of the Exercise is proven analogously to the product series representation of the zeta function The matrix is after a short computation −ν −p =ε 1+ p−1 e ν=0 2πinν p = if n ≡ else mod p , The exercise claims that the development coefficients of the normalized Eisenstein series are satisfying a relation of the shape a(pn) + pk−1 a(n/p) = λ(p)a(n) By Exercise such a relation occurs indeed, and the involved eigenvalue is λ(p) = a(p) If p and n are relatively prime, then we get relation (a), else we have to use also relation (b) 516 VIII Solutions to the Exercises Apply Exercise first for n = 1, then for an arbitrary n The convergence follows from the estimation |a(n)| ≤ C nk−1 (Exercise in VI.4) From the recursion formula for a(pν ) in Exercise we obtain after multiplicative expansion ∞ − a(p)x + pk−1 x2 a(pn )xn 1+ =1 n=1 The involves power series converges for |x| < The product decomposition D(s) = Dp (s) follows from the relation a(nm) = a(n)a(m) for relatively prime integers n, m ≥ by termwise multiplication In full analogy with the case of the product series for the zeta function one has to prove that this formal multiplication of an infinite product is allowed Directly from the definition (Exercise 4) of T (p), and passing to the limit y → ∞, we see that T (p) transforms cusp forms into cusp forms 10 From the formula for T (p) (Exercise 4) it follows f (z) ≤ pk−1 |f (pz)| + p p−1 f ν=0 z+ν p and from this we get k k p−1 |g(z)| ≤ p −1 |g(pz)| + p −1 g ν=0 z+ν p k This gives |g(z)| ≤ p −1 (1 + p)m, and the required estimation If the eigenform f does not vanish identically, then m = λ(p)m This gives the searched estimation for λ(p) If there are given two non-cusp (modular) forms, then a suitable non-trivial linear combination delivers a cusp form If the two non-cusp forms are eigenforms of an operator T (p), then the linearly combined cusp form is an eigenform for the same eigenvalue + pk−1 Hence it must vanish, since for any k ≥ 4, and k any p we have p −1 < + pk−1 Exercises in Sect VII.3 The series lies in the space 1, 2k, (−1)k By Theorem VII.3.4 this space is isomorphic to 1, 2k, (−1)k , which is the space of modular forms of weight 2k In case of k = this space is trivial, in the cases k = 2, 3, it is 1-dimensional being generated by the Eisenstein series Now one can make use of Exercise in VII.2 The proof follows the lines in Exercise Near VII.3.9 one should also use a characterization of ϑk , k < 8, e.g as it can be obtained from Exercise in VI.6 The discriminant is up to constant factors the unique modular form of weight 12, whose development coefficients are of the order O(n11 ) VIII.7 Solutions to the Exercises of Chapter VII 517 In the first series the subseries where the index runs from to ∞, and respectively from −1 to −∞ coincide as it can be seen by the substitution n → −1−n The terms with even n =: 2m of the second series are giving rise to the terms from to ∞ of the third series Correspondingly, the terms with odd n =: 2m+1 are giving rise to the terms of the third series from −1 to −∞ The representation of f as a derivative of the Jacobi theta function evaluated in w = 1/4 is clear from the third formula for f Now apply the differentiation with respect to w in the theta transformation formula, and then specialize w = 1/4 We write f as ∞ (−1)n (2n + 1)e f (z) = 2πi(2n+1)2 n=0 and obtain the associated Dirichlet series in the form ∞ D(s) = (−1)n (2n + 1)(2n + 1)−2s = n=0 ∞ (−1)n (2n + 1)1−2s = L(2s − 1) n=0 The functional equation for D ∈ {8, 3/2, 1} in conformity with VII.3.2 is identical with the searched functional equation for L The functional equation of the Riemann zeta function, and the functional equation for L(s) from Exercise 5, connected with Legendre’s relation IV.1.12 for the gamma function lead to the wanted functional equation for ζ(s)L(s) The normalizing factor are obtained by passing to the limit σ → ∞ Exercises in Sect VII.4 We better define µ(n) by the required formulas The convergence of the Dirichlet series with coefficients µ(n) for σ > is clear Because of the uniqueness of the development as a Dirichlet series the claim of the Exercise is equivalent to ∞ ∞ µ(n) · =1 s n ns n=1 n=1 N This means, that for C(N ) := n=1 µ(n) we have C(1) = 1, and C(N ) = for all N > Because of the obvious relations µ(nm) = µ(n)µ(m) and C(nm) = C(n)C(m) for relatively prime values of m, n we can restrict ourselves to N = pm In case of m > the defining sum for C(N ) has only two nonvanishing terms, and −1 (first summands) If we know the formula for two intervals ]x, y] and ]y, z], then we also know it for ]x, z] It is then enough to prove the formula for those intervals, which not contain any natural numbers in their interior Then the function A(t) is constant in the interior of such an interval, and the claim is easy to check now In Sect VII.4we showed, that the first two formulas are equivalent, and the third one follows follows from them An attentive examination of the proof also gives the converse The convergence of the Dirichlet series with the coefficients ϕ(n) for σ > follows from the trivial estimation ϕ(n) ≤ n The claimed identity 518 VIII Solutions to the Exercises ∞ ∞ nn−s = n=1 ∞ n−s n=1 ϕ(n)n−s n=1 is equivalent to the well-known relation ϕ(d) = n d|n As in the hint, let us suppose the converse A formal computation, which has to be of course rigorously argumented, gives −νs log − p−s = p−s + p ν p p ν≥2 The double series converges even in the region σ > 1/2, as it can be seen by a comparison with the zeta function The first series in the R.H.S is dominated by the series p−1 , which converges by assumption The whole R.H.S remains thus bounded when we pas to the limit s Since the L.H.S is a logarithm of the zeta function, we infer that the zeta function ζ(s) stays bounded for s 1, a contradiction In the region σ > we have the identity ∞ − 21−s ζ(s) = n=1 (−1)n−1 ns By Leibniz’ criterion of convergence for alternated series, the R.H.S is convergent for any real σ > From Exercise of VII.2 we deduce that the R.H.S is even an analytic function in the region σ > From the principle of analytic continuation we deduce this identity in this whole half-plane The alternated series is always positive in the interval ]0, 1[, the factor preceding the zeta function in the L.H.S is negative From the Prime Number Theorem we first easily deduce lim x→∞ log π(x) =1 log x Substituting instead of x the nth prime number pn we deduce from π(pn ) = n lim n→∞ n log n =1 pn Conversely, let us assume that this relation is satisfied For any fixed given x > we then consider the largest prime number pn smaller then x, i.e pn ≤ x < pn+1 From the assumption it easily follows x x lim = lim =1 (∗) x→∞ n log n x→∞ π(x) log π(x) Taking logarithms this implies lim log π(x) + log log π(x) − log x = x→∞ We divide by log π(x) and obtain lim x→∞ log x =1, log π(x) and also using (∗) we get the Prime Number Theorem VIII.7 Solutions to the Exercises of Chapter VII 519 Exercises in Sect VII.5 The Laurent series exists by the general Proposition III.5.2 It remains to is the Euler-Mascheroni constant γ show that γ := lim ζ(s) − s→1 s−1 (page 204) By Lemma VII.5.2 there holds γ= 1 − F (1) = − 2 ∞ β(t) dt t2 The claim now follows from the formula N n=1 1 − log N = + − n 2N N β(t) dt t2 after passing to the limit with N → ∞ The applied formula can be proven by partial integration (compare with the proof of VII.5.2) Both formulas are clear in the convergence domain σ > The series (−1)n−1 n−s converges by Leibniz’ criterion for alternated series first for all real s > By Exercise in VII.2 we obtain convergence in the half-plane σ > 0, and the represented function is analytic in this domain The function ζ(s) can be thus analytically extended in the region σ > 0, excepting the zeros of − 21−s from it Using Q(s) we analogously can deduce the analytic continuation in the region σ > 0, excepting the zeros of − 31−s from it The only common zero is s = The residue is ∞ (−1)n−1 s−1 lim s→1 − 21−s n n=1 The value of the alternated series is known to be log 2, the whole limit is thus equal to From the functional equation written in symmetric format π− 1−s Γ s 1−s s ζ(1 − s) = π − Γ ζ(s) , 2 and from the Completion Formula for the gamma function written as Γ 1+s 1−s Γ 2 = π sin πs + π we deduce ζ(1 − s) = Γ s + −s− 12 πs s π Γ π ζ(s) sin + 2 2 The claim now follows from the Doubling Formula IV.1.12 (a) The pole of ζ(s) is compensated by the prefactor s − 1, the pole of Γ (s/2) in by the prefactor s, and the remained poles by the zeros of the zeta function (Exercise 3) (b) This is the functional equation of the zeta function, if we observe that the prefactor s(s − 1) satisfies the same functional equation (c) Show Φ(s) = Φ(s), and use the functional equation ... the Prime Number Theorem with a weak estimate for the error term In all chapters there are numerous exercises, easy ones at the beginning, but with increasing chapter number there will also be... simple dependency rules for variables expressed by variable operations If one applies these dependency rules in an extended manner by associating complex values to the variables referred to by these... 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