SOME ASPECTS OF ANALYTIC NUMBER THEORY: PARITY, TRANSCENDENCE, AND MULTIPLICATIVE FUNCTIONS

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1 SOME ASPECTS OF ANALYTIC NUMBER THEORY: PARITY, TRANSCENDENCE, AND MULTIPLICATIVE FUNCTIONS by Michael J. Coons B.A., The University of Montana, 2003 M.S., Baylor University, 2005 a thesis submitted in artial fulfillment of the requirements for the degree of Doctor of Philosohy in the Deartment of Mathematics c Michael J. Coons 2009 SIMON FRASER UNIVERSITY Sring 2009 All rights reserved. This work may not be reroduced in whole or in art, by hotocoy or other means, without the ermission of the author.

2 APPROVAL Name: Degree: Title of thesis: Michael J. Coons Doctor of Philosohy Some asects of analytic number theory: arity, transcendence, and multilicative functions Examining Committee: Dr. Steven Ruuth Chair Dr. Peter B. Borwein Senior Suervisor, Simon Fraser University Dr. Stehen K. K. Choi Co Suervisor, Simon Fraser University Dr. Nils Bruin Suervisory Committee, Simon Fraser University Dr. Karen Yeats Internal Examiner, Simon Fraser University Dr. Michael A. Bennett External Examiner, University of British Columbia Date Aroved: Aril 1, 2009 ii

3 Abstract Questions on arities lay a central role in analytic number theory. Proerties of the artial sums of arities are intimate to both the rime number theorem and the Riemann hyothesis. This thesis focuses on investigations of Liouville s arity function and related comletely multilicative arity functions. We give results about the artial sums of arities as well as transcendence of functions and numbers associated to arities. For examle, we show that the generating function of Liouville s arity function is transcendental over the ring of rational functions with coefficients from a finite field. Within the course of investigation, relationshis to finite automata are also discussed. iii

4 iv In memory of Michael and William

5 A boundary between arithmetic and analytic areas of mathematics cannot be drawn. Neuer Beweis der Gleichung µ(k) k=1 k = 0, Edmund Landau If you want to climb the Matterhorn you might first wish to go to Zermatt where those who have tried are buried. A note to a student working on the Riemann hyothesis, György Pólya v

6 Acknowledgments I thank my advisors Peter B. Borwein and Stehen K. K. Choi for their encouragement, comments, criticisms, and suort. They have been, and continue to be, significant influences on my life as a mathematician. I thank the SFU UBC number theory grou and the IRMACS centre. The active community of number theorists and the combined SFU UBC research seminar rovide an amazing environment for research and collaboration. This community is a wonderful lace for a motivated student, and for this exerience, I thank everyone involved. I thank my arents, Robin and Merrily, for their continued suort and encouragement. Lastly, I thank my wife, Alissa, for heling me navigate through many years of study and research. She has suorted me every day, both emotionally and ractically, through three degrees and four countries. Her atience, like her love, is without bounds. vi

7 Contents Aroval Abstract Dedication Quotation Acknowledgments Contents List of Figures Preface ii iii iv v vi vii ix x 1 Introduction Primes and arity The rime number theorem A useful equivalence A density residue theorem Proofs of Theorems 1.11 and Generalized Liouville functions Introduction Proerties of L A (x) One question twice vii

8 2.4 The functions λ (n) A bound for L (n) Mahler s method via two examles Stern s diatomic sequence Transcendence of A(z) Transcendence of A(x, z) Transcendence of F (q) and G(q) Golomb s series A general transcendence theorem The series G k (z) and F k (z) Irrationality and transcendence Formal ower series Values of ower series The Liouville function for rimes 2 modulo The Gaussian Liouville function Transcendence related to character like functions (Non)Automaticity Automaticity (Non)Automaticity of arithmetic functions Dirichlet series and (non)automaticity Dirichlet series and (non)regularity Possible future directions Sums of multilicative functions Algebraic character of generating functions Transcendence and functional equations Transcendental values of series Correlation and diversity A Proof of Mahler s Theorem 85 Bibliograhy 88 viii

9 List of Figures 4.1 The 2 automaton that roduces the sequence G ix

10 Preface As the title would hoefully lead eole to believe, this thesis is dedicated to develoing some of the asects of analytic number theory dealing with arity, transcendence, and multilicative functions. By arity, we mean just that, even or odd. We focus on the arity of the number of rime divisors of an integer. This idea is embodied, or enfunctioned, in the Liouville λ function, which given an integer n, is defined to be 1 if the number of rime divisors of n, counting multilicity, is even, and 1 if odd. By design, λ is comletely multilicative; that is, for all m, n N we have λ(mn) = λ(m)λ(n). Liouville s function is related to some very interesting theorems from rime number theory. Furthermore, the rime number theorem is equivalent to the statement that n x λ(n) = o(x), and the Riemann hyothesis is equivalent to the statement that for every ε > 0, we have n x λ(n) = O(x1/2+ε ). Since an imrovement on the asymtotic behaviour of n x λ(n) is beyond our gras, we dwell uon some questions that we can answer, like what about the artial sums of functions that are similar to Liouville s function?, where similar will be determined later. We also address questions concerning the algebraic character of ower series n 1 f(n)zn and secial values of these series, where f is one of these similar functions. To this end, this thesis is organized as follows. Chater 1 contains an introduction to the theory surrounding Liouville s function by roviding a link to the classical theory of the distribution of rimes. Included in this chater is a new roof of a theorem of Landau and von Mangoldt, which states that the rime number theorem is equivalent to n 1 = 0. We also give a new roof of the statement n x λ(n) = o(x) by roviding a connection between the asymtotic density of a sequence and the residue of the zeta function associated to this sequence. λ(n) n x

11 In Chater 2 we broaden our focus by considering generalized versions of λ. In articular, define the Liouville function for A, a subset of the rimes P, by λ A (n) = ( 1) Ω A(n) where Ω A (n) is the number of rime factors of n coming from A counting multilicity. For the traditional Liouville function, A is the set of all rimes. Denote L A (n) := L A (n) λ A (k) and R A := lim n n. k n Granville and Soundararajan [51] have shown that for every α [0, 1] there is an A P such that R A = α. Given certain restrictions on A, asymtotic estimates for L A (n) are also given. For character like functions λ (λ agrees with a Dirichlet character χ when χ(n) 0) exact values and asymtotics are given; in articular λ (k) log n. k n Within the course of discussion, the ratio ϕ(n)/σ(n) is considered. Chater 3 contains an excursion into Mahler s method of roving transcendence which will be used heavily in Chater 4. This method is used to rove the transcendence of ower series which satisfy certain functional equations. This chater is divided into two sections which deal with two canonically different tyes of functional equations. In the first section of this chater, we give various transcendence results regarding functions related to the Stern sequence. In articular, we rove that the generating function of the Stern sequence is transcendental. Transcendence results are also roven for the generating function of the Stern olynomials and for ower series whose coefficients arise from some secial subsequences of the Stern sequence. In the second section, we rove that a non zero ower series F (z) C[[z]] satisfying F (z d ) = F (z) + A(z) B(z), where d 2, A(z), B(z) C[z] with A(z) 0 and deg A(z), deg B(z) < d is transcendental over C(z). Using this result and Mahler s Theorem, we extend results of Golomb and Schwarz on transcendental values of certain ower series. In articular, we rove that for all k 2 the series G k (z) := n 0 zkn (1 z kn ) 1 is transcendental for all algebraic numbers z with z < 1. We give a similar result for F k (z) := n 0 zkn (1 + z kn ) 1. In Chater 4 we give a new roof of Fatou s theorem: if an algebraic function has a ower series exansion with bounded integer coefficients, then it must be a rational function. This xi

12 result is used to show that for any non trivial comletely multilicative function f : N { 1, 1}, the series n 1 f(n)zn is transcendental over Z(z). For examle, n 1 λ(n)zn is transcendental over Z(z), where λ is Liouville s function. The transcendence of n 1 µ(n)zn is also roved. We continue by considering values of similar series. The Liouville number, denoted l, is the binary number l := , where the nth bit is given by 1 2 (1 + λ(n)); here, as before, λ is Liouville s function. Presumably the Liouville number is transcendental, though at resent, we know of no methods to aroach roof. Similarly, define the Gaussian Liouville number by γ := where the nth bit reflects the arity of the number of rational Gaussian rimes dividing n, 1 for even and 0 for odd. In the second art of this chater, using the methods develoed in Chater 3, we rove that the Gaussian Liouville number and its relatives are transcendental. One such relative is the number k 0 2 3k 2 3k k + 1 = , where the nth bit is determined by the arity of the number of rime divisors that are equivalent to 2 modulo 3. In Chater 5, using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of rime numbers, we rove that λ(n) is not k automatic for any k > 2. This yields that n 1 λ(n)xn F [[X]] is transcendental over F (X) for any rime > 2. Similar results are roven (or reroven) for many common number theoretic functions, including ϕ, µ, Ω, ω, ρ, and others. Throughout Chaters 4 and 5, relationshis to finite automata are discussed. The sixth and final chater of this thesis contains a collection of questions and conjectures for further study. All of the results of this thesis have been ublished or submitted for ublication. We have taken without hesitation from articles to which the author has been a major contributor ([13], [16], and [15]) or the sole author ([28], [29], [30], and [31]). xii

13 Chater 1 Introduction Introduisons maintenant une fonction numérique nouvelle λ(m), dont la valeur soit 1 ou 1, suivant que le nombre total des facteurs remiers, égaux ou inégaux, de m est air ou imair. En d autres termes, soit λ(1) = 1, et généralement, our m décomosé en facteurs remiers sous la forme m = a α b β... c γ, soit λ(m) = ( 1) α+β+...+γ. Cette fonction λ(m), rise isolément ou jointe à celles dont il a été question lus haut, donnera lieu à des théorèmes curieux. [72] 1.1 Primes and arity Recall that the Liouville λ function is the unique comletely multilicative function for which λ() = 1 for all rimes. This function was already considered by Euler, 130 years before Liouville introduced the λ notation. In 1737, Euler stated the following theorems. Theorem 1.1 (Euler [47]). If we take to infinity the continuation of these fractions , where the numerators are all the rime numbers and the denominators are the numerators less one unit, the result is the same as the sum of the series which is certainly infinity

14 CHAPTER 1. INTRODUCTION 2 Theorem 1.2 (Euler [47]). If we assign a sign to all the rime numbers and comosite numbers are assigned the sign that corresond to them according to the rule of signs in the roduct and with all the numbers we form the series will have, once infinitely continued, sum 0. In modern language, these theorems translate as follows. Theorem 1.3. In some infinite sense, one has that and this series diverges. rime ( 1 1 ) 1 = 1 n, n 1 Theorem 1.4. Let n = k 1 1 kr r be the rime factorization of n (n 2), Ω(n) = r j=1 k j, and λ(n) = ( 1) Ω(n) (using the convention that Ω(1) = 0). In some infinite sense n 1 λ(n) n = 0. The words in some infinite sense are very imortant to the interretations of these theorems. Indeed, as we will see later, one version of Theorem 1.4 is quite trivial and another is equivalent to the rime number theorem. We give modern roofs of both versions later in this chater. Theorem 1.1 introduces us to a very fundamental discovery in the theory of numbers: the zeta function with roduct formula. Although this was given by Euler (1737) many years before Riemann (1859), the zeta function is usually attributed to the latter, and the roduct formula to the former. In modern notation, we denote by ζ(s), the Riemann zeta function as a function of a comlex variable, which for Rs > 1 we have the reresentation, 1 ζ(s) = n s = n 1 where the roduct is taken over all rimes. ( 1 1 s ) 1, (1.1) Much is known about ζ(s). First we need to be able to view this function in a larger sense, in the whole comlex lane. The standard way to analytically continue ζ(s) is to

15 CHAPTER 1. INTRODUCTION 3 begin with continuing ζ(s) to Rs > 0 and then use a functional equation to comlete the continuation to all of C excet for the oint s = 1. For Rs > 1 we have ζ(s) = 1 n s = ( ) 1 n n s 1 (n + 1) s = s n+1 n x s 1 dx. n 1 n 1 n 1 n Recall that x = [x] + {x}, where [x] and {x} are the integer and fractional arts of x, resectively. Since [x] is always the constant n for any x in the interval [n, n + 1), we have ζ(s) = s n 1 Writing [x] = x {x} we have ζ(s) = s n+1 n 1 = s s 1 s [x]x s 1 dx = s x s dx s [x]x s 1 dx. {x}x s 1 dx {x}x s 1 dx. We now observe that since 0 {x} < 1, the imroer integral in the last equation converges when Rs > 0 because the integral 1 x Rs 1 dx converges. Thus this integral defines an analytic function of s in the region Rs > 0. Therefore the meromorhic function on the right-hand side of the last equation gives an analytic continuation of ζ(s) to the region Rs > 0, and the s s 1 term gives the simle ole of ζ(s) at s = 1 with residue 1. We note the definition of the Γ-function. Definition 1.5. For Rs > 0, Γ(s) = 0 t s 1 e t dt. (1.2) For s C \ Z the general Γ-function is given by 1 Γ(s) = ( t) s 1 e t dt, (1.3) 2i sin(πs) where the contour C is oriented counter clockwise and contains the nonnegative real axis. The functions Γ(s) and ζ(s) are related via a functional equation which comletes the analytic continuation of ζ(s) to all of C with the excetion of s = 1. Theorem 1.6 (Riemann [85]). The function ζ(s) satisfies the functional equation ( π s s ) ( ) 2 Γ ζ(s) = π 1 s 1 s 2 Γ ζ(1 s). 2 2 C

16 CHAPTER 1. INTRODUCTION 4 As a function of a comlex variable, ζ(s) is analytic everywhere excet at s = 1. Now consider the zeros of ζ(s), and let us focus on the line Rs = 1 away from the oint s = 1; that is, the line 1 + it for t 0. Since ζ(s) is analytic here, let us suose that there is a zero on this line of order r 1. Using Taylor series, we have that ζ(1 + ε + it) cε r for ε sufficiently small. Since ζ(s) has a ole of order 1 at s = 1, we know that ζ(1 + ε) 1 ε. Since We continue in the standard manner, using Mertens simle identity cos(θ) + cos(2θ) 0. R log ζ(σ + it) = relacing t by 0, t, 2t in the above, one has that so that for all real σ > 1 and t 0, n 1 cos(t log n ) n nσ, 3 log ζ(σ) + 4R log ζ(σ + it) + R log ζ(σ + 2it) 0, ζ 3 (σ) ζ 4 (σ + it)ζ(σ + 2it) 1. Substituting our ε estimates in this equation we have c 4 ε 4r 1 ζ(1 + ε + 2it) 1. Taking the limit as ε 0 imlies that ζ(s) has a ole of order 4r 1 1 at s = 1 + 2it, contradicting the fact that ζ(s) is analytic there. Hence we have shown Theorem 1.7 (Hadamard and de la Vallée Poussin, 1896). ζ(s) 0 on the line Rs = The rime number theorem The rime number theorem states that lim x π(x) x/ log x = 1 (1.4) where π(x) denotes the number of rimes less than or equal to x. This was Gauss original formulation, which was roved indeendently by Hadamard [53] and de la Vallée Poussin [32] in They roved this by showing that ζ(s) 0 in the region Rs 1, where ζ(s) is the Riemann zeta function. Indeed, one has that

17 CHAPTER 1. INTRODUCTION 5 Theorem 1.8 (Hadamard [53], de la Vallée Poussin [32]). The rime number theorem is equivalent to the non vanishing of ζ(s) in the region Rs 1. One may also read the rime number theorem, as given by Landau, in the following way: asymtotically there is an equal robability that a given number is the roduct of an even or an odd number of rimes, with multile factors counted with multilicity [67,. 630]. To formalize this statement, consider the following theorem of von Mangoldt. But first, recall that the Möbius function µ : N { 1, 0, 1} is given by 1 n = 1, µ(n) := 0 if k 2 n for some k 2, ( 1) r n = 1 2 r. Theorem 1.9 (von Mangoldt [94]). The rime number theorem imlies that n 1 µ(n) n = 0. Landau [65] gave a new roof of von Mangoldt s result, again using the rime number theorem, and also roved the converse of Theorem 1.9 [66]. Included in these works, he showed that Theorem 1.10 (Landau [65, 66]). The rime number theorem gives µ(n) = o(x). (1.5) n x In his Handbuch [67], Landau gave roofs of these theorems with the Liouville function in lace of the Möbius function. The traditional way to rove the rime number theorem is via Theorem 1.8. The statements in Theorems 1.9 and 1.10 are much less widely known, though they are of a more intuitive nature. For the remainder of this introduction, we rovide new roofs of the λ analogues of Theorems 1.8 and 1.9. More formally, we rove the following theorems. Theorem The following are equivalent: (i) ζ(s) 0 when Rs 1, (ii) n 1 λ(n) n = 0.

18 CHAPTER 1. INTRODUCTION 6 Theorem Let λ denote Liouville s function. Then n x λ(n) = o(x). To emhasize Landau s quote (see age v of this thesis), the theorems, roofs, and methods contained in this chater are intended to highlight the rich interlay between the arithmetic and analytic areas of mathematics A useful equivalence Let A be a subset of N and denote by A(n) the number of elements in A that are less than or equal to n. When it exists, the asymtotic density of A in N, denoted d(a), is given by A(n) d(a) := lim n n. For each ε i {, +} denote by L εi the set L εi := {n N : λ(n) = ε i 1}. We make use of the following equivalence. Lemma n x λ(n) = o(x) if and only if d(l +) = d(l ) = 1 2. Proof. This statement is easily realized by the fact that L + (N) L (N) 1 d(l + ) d(l ) = lim = lim N N N N λ(n). (1.6) If d(l + ) = d(l ) = 1 2, using (1.6), lim n 1 n k n λ(k) = 0 trivially. 1 Conversely, if lim N N n N λ(n) = 0, again aealing to (1.6), it must be the case that d(l + ) = d(l ). Noting that L + (n) + L (n) lim = 1, n n requires the common value of d(l + ) and d(l ) to be 1 2. To establish Theorem 1.12, we rove an amazingly simle link between the concet of density in elementary number theory and the asymtotic behavior of certain zeta functions. n N A density residue theorem For A a subset of N we define the zeta function associated to A, denoted ζ A (s), as ζ A (s) := a A where s is taken to be in the half lane of convergence. Using these definitions we have the following theorem. 1 a s,

19 CHAPTER 1. INTRODUCTION 7 Theorem Let A be a subset of N and s = 1 be the right-most ole of ζ A (s). If s = 1 is a simle ole of ζ A (s) and ζ A (s) can be analytically continued to a region R which contains the half lane R(s) 1 (s 1), then d(a) exists and is equal to Res s=1 {ζ A(s)}. Proof. Following [7,. 243 Lemma 4], we define F (x) := 1 2πi c+i c i x s ds s = 1 if x > if x = 1 0 if 0 < x < 1. (c > 0) Sending x x/a and summing over all a A with a x gives, for ε > 0 some arbitrarily small quantity, we have A(x) = 1 2πi 1+ε+i 1+ε i a x 1 a A 1 ds xs + c F (1), (1.7) as s where c = 1 if x A and c = 0 if x / A. In either case, clearly c F (1) = o(x). Since F (x/a) = 0 when x < a, we may extend the sum in (1.7) to all of A. Hence A(x) = 1 2πi 1+ε+i 1+ε i ζ A (s) x s ds s + c F (1). Since ζ A (s) is analytically continuable to a region R containing R(s) 1 (s 1) and the right-most ole of ζ A (s) is simle, and occurs at s = 1, we gain A(x) = x Res {ζ A(s)} + 1 ζ A (s) x s ds s=1 2πi s + c F (1), so that A(x) x R = Res {ζ A(s)} + 1 s=1 x 1 ζ A (s) x s ds 2πi R s + 1 c F (1), (1.8) x where R denotes the boundary of R. Since R is a region of analyticity of a function, R is oen, and so R contains the right half lane R(s) 1; thus the integral in (1.8) is o(x). To make this exlicit, one may take the boundary of this region to be the contour { C := 1 f(t) } 2 + it : t R, where f(t) is the distance from the oint 1 + it to the nearest ole of ζ A (s) in the right half lane Rs < 1. Since ζ A (s) can be analytically continued to a region R which contains the half lane R(s) 1 (s 1), the distance from each oint on the line Rs = 1 to C

20 CHAPTER 1. INTRODUCTION 8 is necessarily ositive, and hence so is the distance to R. Hence we have shown that C bounds a region R which contains the half lane Rs 1. Thus the limit of the right-hand side as x of (1.8) exists and is equal to Res {ζ A(s)}. s=1 Hence the limit of the left-hand side of (1.8) exists and is equal to Res {ζ A(s)}; that is, d(a) s=1 exists and A(x) d(a) = lim = Res x x {ζ A(s)}, s=1 which is the desired result. The roof of Theorem 1.14 is new, though the result is not. Indeed, Theorem 1.14 contains secial cases of both the Wiener Ikehara Theorem [60, 95] and the Halász Wirsing Mean Value Theorem [54, 97], the roofs of which, in full generality, are much more involved than the secial case given above Proofs of Theorems 1.11 and 1.12 Proof of Theorem Noting that (1 z 2 ) = (1 + z)(1 z), using the Euler roduct formula we have for Rs > 1 n 1 λ(n) n s = ( 1 λ() s ) 1 = ( s ) 1 = Since ζ(s) has a ole at s = 1, and converges at s = 2, we have that ζ(2s) lim s 1 + ζ(s) = 0. λ(n) n s ( ) ( 2s ) 1 = ζ(2s) 1 1 ζ(s). s To construct an analytic continuation of n 1 to the region Rs 1, we define ζ(2s) ζ(s) in the region Rs 1, s 1, Z(s) := 0 on the line s = 1. Now if Z(s) is analytic in the region Rs 1 we have found the unique analytic continuation λ(n) n s of n 1 to this region. Note here that Z(s) is analytic in the region Rs 1 if and only if ζ(s) is non vanishing in this region; this gives the equivalence of (i) and (ii) of Theorem 1.11.

21 CHAPTER 1. INTRODUCTION 9 Proof of Theorem Consider the function ζ L+ (s) = n L + n s. For R(s) > 1 we have ζ L+ (s) = n N l(n)n s where l : N {0, 1} is defined by Also for R(s) > 1, ζ L+ (s) = 1 2 n N l(i) := 1 + λ(i) λ(n) n s = 1 ( ζ(s) + ζ(2s) ) 2 ζ(s) = ζ(s)2 + ζ(2s). (1.9) 2 ζ(s) Since ζ(s) is analytically continuable to a meromorhic function on all of C, the relation in (1.9) imlies the same for ζ L+ (s). Again using (1.9), since ζ(s) is non-zero in the region Rs 1, the function ζ L+ (s) has no oles in the region Rs 1, excet at s = 1. Furthermore, { Res ζl+ (s) } = 1 s=1 2 Res {ζ(s)}. s=1 Hence ζ L+ (s) is analytic at s = 1 + it for all real t 0, since at these s, ζ(s) is nonzero and analytic. Thus, the existence of a meromorhic continuation of ζ L+ (s) to all of C, imlies the existence of a region of analyticity of ζ L+ (s) containing the right half lane Rs 1 with the excetion of the ole at s = 1. Using (1.9), the definition of Z(s) in the roof of Theorem 1.11, and the region R described in the receding aragrah, the function ζ L+ (s) satisfies all of the assumtions of Theorem Alying Theorem 1.14 gives both the existence of d(l + ) and the value { d(l + ) = Res ζl+ (s) } = 1 s=1 2 Res {ζ(s)} = 1 s=1 2. An alication of Lemma 1.13 yields n x λ(n) = o(x).

22 Chater 2 Generalized Liouville functions This chater contains results which were found in collaboration with Peter Borwein and Stehen K.K. Choi (see [13] for details). 2.1 Introduction Let Ω(n) be the number of distinct rime factors in n (with multile factors counted multily). Recall that the Liouville λ function is defined by λ(n) := ( 1) Ω(n). So λ(1) = λ(4) = λ(6) = λ(9) = λ(10) = 1 and λ(2) = λ(5) = λ(7) = λ(8) = 1. In articular, λ() = 1 for any rime. It is well-known [55, Section 22.10] that Ω is comletely additive, i.e, Ω(mn) = Ω(m) + Ω(n) for any m and n and hence λ is comletely multilicative, i.e., λ(mn) = λ(m)λ(n) for all m, n N. It is interesting to note that on the set of square-free ositive integers λ(n) = µ(n), where µ is the Möbius function. In this resect, the Liouville λ function can be thought of as a modification of the Möbius function. Similar to the Möbius function, many investigations surrounding the λ function concern the summatory function of initial values of λ; that is, the sum L(x) := λ(n). n x Historically, this function has been studied by many mathematicians, including Liouville, Landau, Pólya, and Turán. Recent attention to the summatory function of the Möbius 10

23 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 11 function has been given by Ng [80, 81]. Larger classes of comletely multilicative functions have been studied by Granville and Soundararajan [50, 51, 52]. One of the most imortant questions is that of the asymtotic order of L(x); more formally, the question is to determine the smallest value of ϑ for which L(x) lim x x ϑ = 0. It is known that the value of ϑ = 1 is given by the rime number theorem [65, 66] and that ϑ = 1 2 +ε for any arbitrarily small ositive constant ε is equivalent to the Riemann hyothesis [14]. The value of ε is best ossible, as lim su x L(x)/ x > ; see Borwein, Ferguson, and Mossinghoff [19]. Indeed, any result asserting a fixed ϑ ( 1 2, 1) would give an exansion of the zero-free region of the Riemann zeta function, ζ(s), to Rs ϑ. Unfortunately, a closed form for L(x) is unknown. This brings us to the motivating question behind the investigation of this chater: are there functions similar to λ, so that the corresonding summatory function does yield a closed form? Throughout this investigation P denotes the set of all rimes. As an analogue to the traditional λ and Ω consider the following definition. Definition 2.1. Define the Liouville function for A P by λ A (n) = ( 1) Ω A(n) where Ω A (n) is the number of rime factors of n, counting multilicity, coming from A. The set of all of these functions is denoted F({ 1, 1}); this notation is introduced by Granville and Soundararajan in [51]. Alternatively, one can define λ A as the comletely multilicative function with λ A () = 1 for each rime A and λ A () = 1 for all / A. Every comletely multilicative function taking only ±1 values is built this way. Also, denote L A (x) := L A (x) λ A (n) and R A := lim n n. n x In this chater, we first consider questions regarding the roerties of the function λ A by studying the limit R A. The structure of R A is determined and it is shown that for each α [0, 1] there is a subset A of rimes such that R A = α. The rest of this chater considers an extended investigation on those functions in F({ 1, 1}) that are character like in nature, meaning that there is a real Dirichlet character χ such that λ A (n) = χ(n) whenever χ(n) 0. Within the course of discussion, the ratio ϕ(n)/σ(n) is considered.

24 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS Proerties of L A (x) Define the generalized Liouville sequence as L A := (λ A (1), λ A (2),...). Theorem 2.2. If A, then the sequence L A is not eventually eriodic. Proof. Towards a contradiction, suose that L A is eventually eriodic, say the sequence is eriodic after the M th term and has eriod k. Now there is an N N such that for all n N, we have nk > M. Since A, ick A. Then λ A (nk) = λ A () λ A (nk) = λ A (nk). But nk nk(mod k), a contradiction to the eventual k eriodicity of L A. Corollary 2.3. If A P is nonemty, then λ A is not a Dirichlet character. Proof. This is a direct consequence of the non eriodicity of L A. To get more acquainted with the sequence L A, we study the artial sums L A (x) of L A, and to study these, we consider the Dirichlet series with coefficients λ A (n). Starting with singleton sets {} of the rimes, a nice relation becomes aarent; for Rs > 1 we have and for sets {, q}, the following identity holds: (1 s ) (1 + s ) ζ(s) = λ {} (n) n s, (2.1) n 1 (1 s )(1 q s ) (1 + s )(1 + q s ) ζ(s) = λ {,q} (n) n s. (2.2) n 1 Since λ A is comletely multilicative, for any subset A of rimes, for Rs > 1 we have L A (s) := λ A (n) n s = λ A ( l ) ls n 1 l 0 = ( 1) l 1 ls ls = ( ) ( ) A l 0 A l 0 A 1 1 s A s = ζ(s) ( ) 1 s 1 + s. (2.3) A

25 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 13 This relation leads us to our next theorem, but first let us recall a iece of notation from the last section. Definition 2.4. For A P denote R A := lim n λ A (1) + λ A (2) λ A (n). n The existence of the limit R A is guaranteed by Wirsing s Theorem. In fact, Wirsing [97] showed more generally that every real multilicative function f with f(n) 1 has a mean value, i.e, the limit 1 lim x x exists. Furthermore, Wintner [96] showed that 1 lim f(n) = x x n x f(n) n x ( 1 + f() ) ( + f(2 ) ) 0 if and only if 1 f() / converges. Otherwise the mean value is zero. This gives the following theorem. Theorem 2.5. For the comletely multilicative function λ A (n), the limit R A exists and 1 A +1 if A R A = 1 <, 0 otherwise. Examle 2.6. For any rime, R {} = (2.4) Let us make some notational comments. Denote by P(P ) the ower set of the set of rimes. Note that = Recall from above that R : P(P ) R is defined by R A := ( 1 2 ). + 1 A It is immediate that R is bounded above by 1 and below by 0, so that we need only consider that R : P(P ) [0, 1]. It is also immediate that R = 1 and R P = 0.

26 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 14 Remark 2.7. For n N, let n be the smallest rime larger than n 3 ; i.e. n := min q>n 3{q P }. Since there is always a rime in the interval (x, x + x 5/8 ] (see [61]), we have n+1 > n for all n N. Let K := { n : n N} = {11, 29, 67, 127, 223, 347, 521, 733, 1009, 1361,...}. Note that so that R K = K n 1 n + 1 > n3 1 n 3 + 1, ( ) 1 > ( n 3 ) n 3 = n 2 Also R K < (11 1)/(11 + 1) = 5/6, so that and R K (0, 1). 2 3 < R K < 5 6, There are some very interesting and imortant examles of sets of rimes A for which R A = 0. Indeed, results of von Mangoldt [94] and Landau [65, 66] give the following equivalence. Theorem 2.8. The rime number theorem is equivalent to R P = 0. We may be a bit more secific regarding the values of R A, for A P(P ). For each α (0, 1), there is a set of rimes A such that R A = ( ) 1 = α. + 1 A This result is a secial case of some general theorems of Granville and Soundararajan [51]. Theorem 2.9 (Granville and Soundararajan [51]). The function R : P(P ) [0, 1] is surjective. That is, for each α [0, 1] there is a set of rimes A such that R A = α. Proof. This follows from Corollary 2 and Theorem 4 (ii) of [51] with S = { 1, 1}, though in this secial case, a much more elementary argument can yield the result. To this end, not first that R P = 0 and R = 1. To rove the statement for the remainder of the values, let α (0, 1). Then since ( lim R {} = lim 1 2 ) = 1, + 1

27 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 15 there is a minimal rime q 1 such that i.e., R {q1 } = ( 1 2 ) > α, q α R {q 1 } = 1 ( 1 2 ) > 1. α q Similarly, for each N N, we may continue in the same fashion, choosing q i > q i 1 (for i = 2... N) minimally, we have Now consider 1 α R {q 1,q 2,...,q N } = 1 α N i=1 1 lim N α R {q 1,q 2,...,q N } = 1 α ( 1 2 ) > 1. q i + 1 i 1 ( 1 2 ), q i + 1 where the q i are chosen as before. Denote A = {q i : i N}. We know that 1 α R A = 1 ( 1 2 ) 1. α q i + 1 We claim that R A = α. To this end, let us suose to the contrary that 1 α R A = 1 ( 1 2 ) > 1. α q i + 1 Note that P \A is infinite (here P is the set of all rimes). As earlier, since ( lim R {} = lim 1 2 ) = 1, + 1 A\P there is a minimal rime q A\P such that 1 α R A R {q} = 1 ( 1 2 ) ( 1 2 ) > 1. α q i + 1 q + 1 i 1 i 1 i 1 Since q is a rime and q / A, there is an i N with q i < q < q i+1. This contradicts that q i+1 was a minimal choice. Hence 1 α R A = 1 α i 1 and there is a set A of rimes such that R A = α. ( 1 2 ) = 1, q i + 1

28 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 16 In fact, let S denote a subset of the unit disk and let F(S) be the class of totally multilicative functions such that f() S for all rimes. Granville and Soundararajan [51] rove very general results concerning both the Euler roduct sectrum Γ θ (S) and the sectrum Γ(S) of the class F(S). The following theorem gives asymtotic formulas for the mean value of λ A if certain conditions on the density of A in P are assumed. Theorem Let A be a subset of rimes and suose x A log = 1 κ 2 log x + O(1) (2.5) with 1 κ 1. If 0 < κ 1, then we have n x λ A (n) n = c κ (log x) κ + O(1) and where In articular, c κ = 1 R A = lim x x λ A (n) = (1 + o(1))c κ κx(log x) κ 1, n x 1 Γ(κ + 1) ( 1 1 ) κ ( 1 λ ) 1 A(). (2.6) c 1 = ( ) 1 A +1 if κ = 1, λ A (n) = 0 if 0 < κ < 1. n x Furthermore, L A (s) = n 1 that is, λ A (n) n s has a ole of order κ at s = 1 with residue c κ Γ(κ + 1); L A (s) = c κγ(κ + 1) (s 1) κ + ψ(s), Rs > 1, for some function ψ(s) analytic in the region Rs 0. If 1 κ < 0, then L A (s) has zero of order κ at s = 1 and L A (s) = ζ(2s) c κ Γ( κ + 1) (s 1) κ (1 + ϕ(s))

29 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 17 for some function ϕ(s) analytic in the region Rs 1 and hence L A (1) = n 1 λ A (n) n = 0 and 1 R A = lim x x λ A (n) = 0. If κ = 0, then L A (s) has neither ole nor zero at s = 1. In articular, we have n 1 λ A (n) n n x = α 0 and 1 R A = lim x x λ A (n) = 0. n x The roof of Theorem 2.10 requires the following result. Theorem 2.11 (Wirsing [97]). Suose f is a comletely multilicative function which satisfies and Λ(n)f(n) = κ log x + O(1) (2.7) n x f(n) log x (2.8) n x with 0 κ 1 where Λ(n) is the von Mangoldt function. Then we have f(n) = c f (log x) κ + O(1) (2.9) n x where c f := 1 Γ(κ + 1) where Γ(κ) is the Gamma function. ( 1 1 ) κ ( ) 1 1 f() Proof of Theorem Suose first that 0 < κ 1. We choose f(n) = λ A(n) n Theorem. Since n x Λ(n) n = x log + O(1) = log x + O(1), (2.10) in Wirsing s

30 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 18 we have n x Λ(n) λ A(n) n = x = x = x log λ A() log λ A() log λ A() + O l x l 2 + O n x + O(1). log l Λ(n) n x log On the other hand, from (2.5) we have x log λ A() = x log 2 x A = κ log x + O(1). log Hence and condition (2.7) is satisfied. n x Λ(n) λ A(n) n It then follows from (2.9) and (2.6) that From (2.5), L A (s + 1) = n 1 n x λ A (n) n λ A (n) n s+1 = = = κ log x + O(1) = c κ (log x) κ + O(1). 1 1 = c κ κ y s d n y λ A (n) n y s d (c κ (log y) κ + O(1)) 1 (log y) κ 1 y s+1 dy + = c κ Γ(κ + 1)s κ + ψ(s) 1 y s do(1) for Rs > 0, because 1 (log y) κ 1 y s+1 dy = Γ(κ)s κ.

31 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 19 Here ψ(s) is an analytic function on Rs 0. Therefore, L A (s) has a ole at s = 1 of order 0 < κ 1. Now from a generalization of the Wiener-Ikehara theorem [9, Theorem 7.7], we have λ A (n) = (1 + o(1))c κ κx(log x) κ 1 and hence where n x 1 R A = lim x x c 1 = c 1 if κ = 1, λ A (n) = 0 if 0 < κ < 1, n x ( 1 1 ) ( 1 λ ) 1 A() = A ( Denote the comlement of A by A. If 1 κ < 0, then we have L A (s) = λ A (n) n s = ζ(s) ( ) 1 s 1 + s n 1 A = ζ(2s) ( ) 1 + s ζ(s) 1 s = ζ(2s) L A (s) for Rs > 1. Hence, for Rs > 1, we have and From (2.5), we have x A log = x n x A ). L A (s)l A (s) = ζ(2s). (2.11) log x A Λ(n) λ A (n) n log = 1 + κ 2 = κ log x + O(1). log x + O(1) We then aly the above case to L A (s) and deduce that L A (s) has a ole at s = 1 of order κ, then in view of (2.11), L A (s) has a zero at s = 1 of order κ; that is, L A (s) = ζ(2s) c κ Γ( κ + 1) (s 1) κ (1 + ϕ(s)) for some function ϕ(s) analytic on the region Rs 1. In articular, we have L A (1) = n 1 This comletes the roof of Theorem λ A (n) n = 0. (2.12)

32 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 20 Recall that Theorem 2.9 tells us that any α [0, 1] is a mean value of a function in F({ 1, 1}). The functions in F({ 1, 1}) can be ut into two natural classes: those with mean value 0 and those with ositive mean value. Asymtotically, those functions with mean value zero are more interesting, and it is in this class which the Liouville λ function resides, and in that which concerns the rime number theorem and the Riemann hyothesis. We consider an extended examle of such functions in Section 2.4. Before this consideration, we ask some questions about those functions f F({ 1, 1, }) with ositive mean value. 2.3 One question twice It is obvious that if α / Q, then R A α for any finite set A P. We also know that if A P is finite, then R A Q. Question For α Q is there a finite subset A of P, such that R A = α? The above question can be osed in a more interesting fashion. Indeed, note that for any finite set of rimes A, we have that R A = A = ϕ() σ() = ϕ(z) σ(z) A where z = A, ϕ is Euler s totient function, and σ is the sum of divisors function. Alternatively, we may view the finite set of rimes A as determined by the square free integer z. In fact, the function f from the set of square free integers to the set of finite subsets of rimes, defined by f(z) = f( 1 2 r ) = { 1, 2,..., r }, (z = 1 2 r ) is bijective, giving a one to one corresondence between these two sets. In this terminology, we ask the question as: Question Is the image of ϕ(z)/σ(z) : {square free integers} Q (0, 1) a surjection? That is, for every rational q (0, 1), is there a square free integer z such that ϕ(z) σ(z) = q? As a start, we have Theorem 2.9, which gives a nice corollary.

33 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 21 Corollary If S is the set of square free integers, then x R : x = ϕ(n k ) = [0, 1]; σ(n k ) lim k {n k } S that is, the set {ϕ(s)/σ(s) : s S} is dense in [0, 1]. Proof. Let α [0, 1] and A be a subset of rimes for which R A = α. If A is finite we are done, so suose A is infinite. Write A = {a 1, a 2, a 3,...} where a i < a i+1 for i = 1, 2, 3,... and define n k = k i=1 a i. The sequence (n k ) satisfies the needed limit. 2.4 The functions λ (n) We now turn our attention to those functions F({ 1, 1}) with mean value 0; in articular, we wish to examine functions for which a sort of Riemann hyothesis holds: functions for which L A (s) = λ A (n) n 1 n has a large zero free region. These are functions for which s n x λ A(n) grows slowly. To this end, let be a rime number. Recall that the Legendre symbol modulo is defined as ( ) q = 1 if q is a quadratic residue modulo, 1 if q is a quadratic non-residue modulo, 0 if q 0 (mod ). Here q is a quadratic residue modulo rovided q x 2 (mod ) for some x 0 (mod ). ( Define the function Ω (n) to be the number of rime factors q, of n with q ) = 1; that is, { ( } Ω (n) = # q : q is a rime, q n, and q ) = 1. Definition The Liouville function for quadratic non-residues modulo is defined as λ (n) := ( 1) Ω(n). The function Ω (n) is comletely additive since it counts rimes with multilicities. Thus λ (n) is comletely multilicative.

34 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 22 Lemma The function λ (n) is the unique comletely multilicative function defined by λ () = 1, and for rimes q by ( ) q λ (q) =. ( Proof. Let q be a rime with q n. Now Ω (q) = 0 or 1 deending on whether q ) = 1 or ( 1, resectively. If q ) = 1, then Ω (q) = 0, and so λ (q) = 1. ( On the other hand, if q ) = 1, then Ω (q) = 1, and so λ (q) = 1. Note that using ( ) the given definition λ () = = 1, so that in either case, we have λ (q) = ( ) q. Hence if n = k m with m, then we have λ ( k m) = ( ) m. (2.13) Similarly, we may define the function Ω (n) to be the number of rime factors q of n ( with q ) = 1; that is, { ( } Ω (n) = # q : q is a rime, q n, and q ) = 1. Analogous to Lemma 2.16 we have the following lemma for λ (n) and theorem relating these two functions to the traditional Liouville λ-function. Lemma The function λ (n) is the unique comletely multilicative function defined by λ () = 1 and for rimes q, as λ (q) = ( ) q. Theorem If λ(n) is the standard Liouville λ function, then where k n, i.e., k n and k+1 n. λ(n) = ( 1) k λ (n) λ (n)

35 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 23 Proof. It is clear that the theorem is true for n = 1. Since all functions involved are comletely multilicative, it suffices to show the equivalence for all rimes. λ(q) = 1 for any rime q. Now if n =, then k = 1 and ( 1) 1 λ () λ () = ( 1) (1) (1) = 1 = λ(). If n = q, then ( ) ( ( )) ( ) q q q ( 1) 0 λ (q) λ 2 (q) = = = 1 = λ(q), and so the theorem is roved. Note that To mirror the relationshi between L and λ, denote by L (n), the summatory function of λ (n); that is, define L (n) := n λ (n). k=1 It is quite immediate that L (n) is not ositive for all n and. To find an examle we need only look at the first few rimes. For = 5 and n = 3, we have L 5 (3) = λ 5 (1) + λ 5 (2) + λ 5 (3) = = 1 < 0. Indeed, the next few theorems are sufficient to show that there is a ositive roortion (at least 1/2) of the rimes for which L (n) < 0 for some n N. For the traditional L(n), it was conjectured by Pólya that L(n) 0 for all n, though this was roven to be a non-trivial statement and ultimately false (see Haselgrove [57]). Theorem Let n = a 0 + a 1 + a a k k be the base exansion of n, where a j {0, 1, 2,..., 1}. Then we have L (n) := n a 0 a 1 λ (l) = λ (l) + λ (l) a k l=1 l=1 l=1 l=1 λ (l). (2.14) Here the sum over l is regarded as emty if a j = 0. Instead of giving a roof of Theorem 2.19 in this secific form, we will rove a more general result to which Theorem 2.19 is a direct corollary. Let χ be a non-rincial Dirichlet

36 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 24 character modulo and for any rime q let 1 if q = q, f(q) := χ(q) if q. (2.15) We extend f to be a comletely multilicative function and get f( l m) = χ(m) (2.16) for l 0 and m. Definition Define N(n, l) to be the number of times that l occurs in the base exansion of n. Theorem For N(n, l) as above n 1 f(j) = N(n, l) χ(m). m l j=1 l=0 Proof. We write the base exansion of n as n = a 0 + a 1 + a a k k (2.17) where 0 a j 1. We then observe that, by writing j = l m with m, n f(j) = j=1 k l=0 n f(j) = j=1 l j k l=0 m n/ l (m,)=1 f( l m). For simlicity, we write A := a 0 + a a l l and B := a l+1 + a l a k k l 1 so that n = A + B l+1 in (2.17). It now follows from (2.16) and (2.17) that n f(j) = j=1 k l=0 m n/ l (m,)=1 χ(m) = k χ(m) = l=0 m A/ l +B k χ(m) l=0 m A/ l because χ() = 0 and a+ m=a+1 χ(m) = 0 for any a. Now since a l A/ l = (a 0 + a a l l )/ l < a l + 1,

37 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 25 we have n f(j) = j=1 This roves the theorem. k 1 χ(m) = N(n, l) χ(m). m a l m l l=0 l=0 In this language, Theorem 2.19 can be stated as follows. Corollary We have L (n) = n 1 λ (j) = N(n, l) ( ) m. (2.18) l=0 m l j=1 As an alication of this theorem consider = 3. Alication The value of L 3 (n) is equal to the number of ones in the base 3 exansion of n. Proof. Since ( 1 3) = 1 and ( 1 3) + ( 2 3) = 0, if n = a0 + a a a k 3 k is the base 3 exansion of n, then the right-hand side of (2.14), or equivalently the right-hand side of (2.18), is equal to N(n, 1). The result then follows from either of Theorem 2.19 or Corollary Note that L 3 (n) = k for the first time when n = k and is never negative. This is in stark contrast to the traditional L(n), which is often negative. Indeed, we may classify all for which L (n) 0 for all n N. Theorem The function L (n) 0 for all n exactly for those odd rimes for which for all 1 k. ( ) 1 + ( ) ( ) k 0 Proof. We first observe from (2.13) that if 0 r <, then r λ (l) = l=1 r l=1 ( ) l.

38 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 26 Let n = a 0 + a a k k be the base exansion of n. From Theorem 2.19, n λ (l) = l=1 = a 0 l=1 a 0 l=1 a 1 λ (l) + λ (l) ( ) l l=1 a 1 + l=1 ( ) l a k l=1 a k l=1 λ (l) ( ) l because all a j are between 0 and 1. The result then follows. Corollary For n N, we have 0 L 3 (n) [log 3 n] + 1. Proof. This follows from Theorem 2.24, Alication 3.21, and the fact that the number of 1 s in the base three exansion of n is [log 3 n] + 1. As a further examle, let = 5. Corollary The value of L 5 (n) is equal to the number of 1 s in the base 5 exansion of n minus the number of 3 s in the base 5 exansion of n. Also for n 1, L 5 (n) [log 5 n] + 1. Recall from above, that L 3 (n) is always nonnegative, but L 5 (n) isn t. Also L 5 (n) = k for the first time when n = k and L 5 (n) = k for the first time when n = k. Remark The reason for secification of the rimes in the receding two corollaries is that, in general, it s not always the case that L (n) [log n] + 1. We now return to our classification of rimes for which L (n) 0 for all n 1. Definition Denote by L +, the set of rimes for which L (n) 0 for all n N. We have found by comutation that the first few values in L + are L + = (3, 7, 11, 23, 31, 47, 59, 71, 79, 83, 103, 131, 151, 167, 191, 199, 239, ). By insection, L + doesn t seem to contain any rimes, with 1 (mod 4). This is not a coincidence, as demonstrated by the following theorem.

39 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 27 Theorem If L +, then 3 (mod 4). Proof. Note that if 1 (mod 4), then ( ) a = ( ) a for all 1 a 1, so that 1 2 ( ) a = 0. a=1 ( ) Consider the case that ( 1)/2 = 1. Then 1 2 a=1 1 ( ) a = 2 1 a=1 ( ) a + ( ) ( 1)/2 = a=1 ( ) a + 1, so that On the other hand, if ( ( 1)/ a=1 ( ) a = 1 < 0. ) = 1, then since ( ) ( 1)/2 = ( ) ( 1)/2+1, we have 1 2 a=1 1 ( ) a = 2 +1 a=1 ( ) a ( ) ( 1)/2 + 1 = a=1 ( ) a + 1, so that a=1 ( ) a = 1 < A bound for L (n) Above we were able to give exact bounds on the function L (n). As exlained in Remark 2.27, this is not always ossible, though an asymtotic bound is easily attained with a few reliminary results. Lemma For all r, n N we have L ( r n) = L (n).

40 CHAPTER 2. GENERALIZED LIOUVILLE FUNCTIONS 28 Proof. For i = 1,..., 1 and k N, λ (k + i) = λ (i). For k N, this relation immediately gives that L ((k + 1) 1) L (k) = 0, since L ( 1) = 0. Thus r n L ( r n) = λ (k) = k=1 r 1 n k=1 The lemma follows immediately. λ (k) = r 1 n k=1 λ ()λ (k) = r 1 n k=1 λ (k) = L ( r 1 n). Theorem The maximum value of L (n) for n < k occurs at n = k σ( k 1 ) with value where σ(n) is the sum of the divisors of n. max L (n) = k max L (n), n< k n< Proof. This follows directly from Lemma Corollary If is an odd rime, then L (n) log n; furthermore, max L (x) log x. n x This last corollary begs the question: what can be said about the growth of n x f(n) for any function f F({ 1, 1})? Presumably this quantity is unbounded for all such f, though this is resently unknown.

41 Chater 3 Mahler s method via two examles Before considering values and ower series of more general functions in F({ 1, 1}), we resent two detailed examles using Mahler s transcendence methods. The roofs here were insired by Dekking s roof of the transcendence of the Thue Morse number [34]. The two examles discussed here concern ower series, F (z) C[[z]], which satisfy two very different tyes of functional equations similar to (k 2) F (z k ) = R(z)F (z) and F (z k ) = F (z) + R(z), where R(z) Z(z). 3.1 Stern s diatomic sequence The Stern sequence, sometimes called Stern s diatomic sequence, (a(n)) n 0 is given by a(0) = 0, a(1) = 1, and when n 1, by a(2n) = a(n) and a(2n + 1) = a(n) + a(n + 1). Proerties of this sequence have been studied by many authors; for references see [36]. The Stern sequence is A in Sloane s list (see htt:// njas/ sequences/a000108). In the article cited above, Dilcher and Stolarsky introduced and studied a olynomial analogue of the Stern sequence, defined by a(0; x) = 0, a(1; x) = 1, and when n 1, by a(2n; x) = a(n; x 2 ) and a(2n + 1; x) = x a(n; x 2 ) + a(n + 1; x 2 ). 29

42 CHAPTER 3. MAHLER S METHOD VIA TWO EXAMPLES 30 We call a(n; x) the nth Stern olynomial. Denote by A(z) and A(x, z) the generating functions of the Stern sequence and Stern olynomials, resectively. In this section, we rove that these generating functions are transcendental. There are some secial subsequences of (a(n)) n 0 of interest. It is known (see Lehmer [70] and Lind [71]) that the maximum value of a(m) in the interval 2 n 2 m 2 n 1 is the nth Fibonacci number F n and that this maximum occurs at m = 1 3 (2n ( 1) n ) and m = 1 3 (5 2n 2 + ( 1) n ). Dilcher and Stolarsky [35] set α n := 1 3 (2n ( 1) n ) (n 0), β n := 1 3 (5 2n 2 + ( 1) n ) (n 2), and define for n 0 and for n 2 f n (q) := a(α n ; q) f n (q) := a(β n ; q). Throughout the aer [35] the authors study roerties of f n and f n, finding functional equations and other such relationshis. They are articularly concerned with the functions F and G defined as follows. Definition 3.1. For comlex q with q < 1 we define F (q) : = lim f 2n(q) = lim f 2n+1 (q) n n = 1 + q + q 2 + q 5 + q 6 + q 8 + q 9 + q 10 + q 21 + q 22 + q 24 +, G(q) : = lim f 2n+1(q) = lim f 2n (q) n n = 1 + q + q 3 + q 4 + q 5 + q 11 + q 12 + q 13 + q 16 + q 17 + q In a remark in [35], Dilcher and Stolarsky ask about the transcendence of F and G but make no conclusions. We resolve this question: these functions are transcendental.