COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1}

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX) COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1} PETER BORWEIN, STEPHEN K.K. CHOI, AND MICHAEL COONS Abstract. 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 (x) : X λ A (n) and R A : lim n L A (n). n It is known that for each α [0, 1] there is an A P such that R A α. Given certain restrictions on the sifting density of A, asymtotic estimates for P λ A(n) can be given. With further restrictions, more can be said. For an odd rime, define the character like function λ as λ (k + i) (i/) for i 1,..., 1 and k 0, and λ () 1 where (i/) is the Legendre symbol (for examle, λ 3 is defined by λ 3 (3k + 1) 1, λ 3 (3k + 2) 1 (k 0) and λ 3 (3) 1) For the artial sums of character like functions we give exact values and asymtotics; in articular we rove the following theorem. Theorem. If is an odd rime, then X max λ (k) log x. k n This result is related to a question of Erdős concerning the existence of bounds for number theoretic functions. Within the course of discussion, the ratio φ(n)/σ(n) is considered. 1. Introduction Let Ω(n) be the number of distinct rime factors in n (with multile factors counted multily). 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 (e.g. See of [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 an extension of the Möbius function. Received by the editors June 13, Mathematics Subject Classification. Primary 11N25; 11N37 Secondary 11A15. Key words and hrases. Liouville Lambda Function, Multilicative Functions. Research suorted in art by grants from NSERC of Canada and MITACS.. 1 c XXXX American Mathematical Society

2 2 PETER BORWEIN, STEPHEN K.K. CHOI, AND MICHAEL COONS 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). Historically, this function has been studied by many mathematicians, including Liouville, Landau, Pólya, and Turán. Recent attention to the summatory function of the Möbius function has been given by Ng [16, 17]. Larger classes of comletely multilicative functions have been studied by Granville and Soundararajan [7, 8, 9]. 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 equivalent to the rime number theorem [14, 15] and that ϑ ε for any arbitrarily small ositive constant ε is equivalent to the Riemann hyothesis [3] (The value of ε is best ossible, as lim su x L(x)/ x > , see Borwein, Ferguson, and Mossinghoff [4]). 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 R(s) ϑ. Unfortunately, a closed form for determining L(x) is unknown. This brings us to the motivating question behind this investigation: are there functions similar to λ, so that the corresonding summatory function does yield a closed form? Throughout this investigation P will denote the set of all rimes. As an analogue to the traditional λ and Ω, 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 coming from A counting multilicity. 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. The class of functions from N to { 1, 1} is denoted F({ 1, 1}) (as in [8]). Also, define L A : L A (n) λ A (n) and R A : lim n n. In this aer, we first consider questions regarding the roerties of the function λ A by studying the function R A. The rest of this aer considers an extended investigation of those functions in F({ 1, 1}) which are character like in nature (meaning that they agree with a real Dirichlet character χ at nonzero values). While these functions are not really direct analogues of λ, much can be said about them. Indeed, we can give exact formulae and shar bounds for their artial sums. Within the course of discussion, the ratio φ(n)/σ(n) is considered. 2. Proerties of L A (x) Define the generalized Liouville sequence as L A : {λ A (1), λ A (2),...}. Theorem 1. The sequence L A is not eventually eriodic.

3 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1} 3 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 1. 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 R(s) > 1 (1) and for sets {, q}, (2) (1 s ) (1 + s ) ζ(s) n1 (1 s )(1 q s ) (1 + s )(1 + q s ) ζ(s) λ {} (n) n s, n1 λ {,q} (n) n s. For any subset A of rimes, since λ A is comletely multilicative, for R(s) > 1 we have λ A (n) L A (s) : n s ( ) λ A ( l ) ls n1 l0 ( ) ( ( 1) l ) 1 ls ls A l0 A l0 A 1 1 s A s ζ(s) 1 s (3) 1 + s. A This relation leads us to our next theorem, but first let us recall a vital iece of notation from the introduction. Definition 1. For A P denote λ A (1) + λ A (2) λ A (n) R A : lim. n n The existence of the limit R A is guaranteed by Wirsing s Theorem. In fact, Wirsing in [20] showed more generally that every real multilicative function f with f(n) 1 has a mean value, i.e, the limit 1 lim f(n) x x exists. Furthermore, in [19] Wintner showed that 1 lim x x f(n) ( 1 + f() + f(2 ) 2 + ) ( 1 1 ) 0 if and only if 1 f() / converges; otherwise the mean value is zero. This gives the following theorem.

4 4 PETER BORWEIN, STEPHEN K.K. CHOI, AND MICHAEL COONS Theorem 2. For the comletely multilicative function λ A (n), the limit R A exists and { 1 A +1 if A (4) R A 1 <, 0 otherwise. Examle 1. For any rime, R {} To be a little more descritive, 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. Remark 1. For an examle of a subset of rimes with mean value in (0, 1), consider the set K of rimes defined by { } K : n P : n min q>n3{q P } for n N. Since there is always a rime in the interval (x, x + x 5/8 ] (see Ingham [12]), these rimes are well defined; that is, n+1 > n for all n N. The first few values give Note that so that K {11, 29, 67, 127, 223, 347, 521, 733, 1009, 1361,...}. n 1 n + 1 > n3 1 n 3 + 1, 1 > + 1 R K K n2 Also R K < (11 1)/(11 + 1) 5/6, so that 2 3 < R K < 5 6, and R K (0, 1). ( n 3 ) 1 n There are some very interesting and imortant examles of sets of rimes A for which R A 0. Indeed, results of von Mangoldt [18] and Landau [14, 15] give the following equivalence. Theorem 3. 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 [8].

5 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1} 5 Theorem 4 (Granville and Soundararajan [8]). 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 [8] with S { 1, 1}. 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 [8] 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 a certain condition on the sifting density of A in P is assumed. Theorem 5. Suose A is a subset of rimes with sifting density 1 κ 2 where 1 κ 1. More recisely, and 1 κ 1. If 0 < κ 1, then where c κ 1 Γ(κ+1) 1 R A lim x x If 1 κ 0, then x A log 1 κ 2 log x + O(1) λ A (n) (1 + o(1))c κ κx(log x) κ 1, κ λ A() ; in articular, { c1 1 A +1 if κ 1, λ A (n) 0 if 0 < κ < 1. 1 R A lim x x λ A (n) 0. Proof. This theorem follows from Wirsing s Theorem (Theorem 1.1 on age 27 of [13]) and a generalization of the Wiener Ikehara Theorem (Theorem 7.7 of [2]). Recall that Theorem 4 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 a secial class of functions with mean value 0 in Section 4. Before this consideration, we ask some questions about those functions f F({ 1, 1, }) with ositive mean value. 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 1. Is there a converse to this; that is, for α Q is there a finite subset A of P, such that R A α?

6 6 PETER BORWEIN, STEPHEN K.K. CHOI, AND MICHAEL COONS 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 2. Is the image of φ(z)/σ(z) : {square free integers} Q (0, 1) a surjection? φ(z) σ(z) That is, for every rational q (0, 1), is there a square free integer z such that q? As a start, Theorem 4 gives a nice corollary. Corollary 2. 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 i1 a i. The sequence (n k ) satisfies the needed limit. 4. The functions λ (n) We now turn our attention to a class of those functions in 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 N n has a large s zero free region; that is, functions for which λ A(n) grows slowly. Indeed, the functions we consider in the following sections have artial sums which grow extremely slow, in fact, logarithmically. Because of this slow growth, they give rise to Dirichlet series which converge in the entire right half lane R(s) > 0. Thus our investigation diverges significantly from functions which are very similar to λ, but focuses on those that yield some very interesting results. 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 ).

7 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1} 7 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 2. The modified Liouville function for quadratic non-residues modulo is defined as λ (n) : ( 1) Ω(n). Analogous to Ω(n), since Ω (n) counts rimes with multilicities, Ω (n) is comletely additive, and so λ (n) is comletely multilicative. This being the case, we may define λ (n) uniquely by its values at rimes. Lemma 1. 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 1, then Ω (q) 1, and so λ (q) 1. In either ( q case, we have 1 λ (q) q. Hence if n k m with m, then we have (5) λ ( k m) 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 1 we have the following lemma for λ (n) and theorem relating these two functions to the traditional Liouville λ-function. Lemma 2. The function λ (n) is the unique comletely multilicative function defined by λ () 1 and for rimes q, as q λ (q). ( m Theorem 6. 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) 1 Note that using the given definition λ() 1.

8 8 PETER BORWEIN, STEPHEN K.K. CHOI, AND MICHAEL COONS 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. Note that 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. To mirror the relationshi between L and λ, denote by L (n), the summatory function of λ (n); that is, define n L (n) : λ (k). It is quite immediate that L (n) is not ositive 2 for all n and. To find an examle we need only look at the first few rimes. For 5 and n 3, we have k1 L 5 (3) λ 5 (1) + λ 5 (2) + λ 5 (3) < 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. Theorem 7. 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 n a 0 a 1 a k (6) L (n) : λ (l) λ (l) + λ (l) λ (l). Here the sum over l is regarded as emty if a j 0. Instead of giving a roof of Theorem 7 in this secific form, we will rove a more general result for which Theorem 7 is a direct corollary. To this end, let χ be a non-rincial Dirichlet character modulo and for any rime q let { 1 if q, (7) f(q) : χ(q) if q. We extend f to be a comletely multilicative function and get (8) f( l m) χ(m) for l 0 and m. Theorem 8. Let N(n, l) be the number of digits l in the base exansion of n. Then n 1 f(j) N(n, l) χ(m). m l j1 l0 2 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 [11]).

9 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1} 9 Proof. We write the base exansion of n as (9) n a 0 + a 1 + a a k k where 0 a j 1. We then observe that, by writing j l m with m, n k n k f(j) f(j) f( l m). j1 For simlicity, we write l0 j1 l j l0 m n/ l (m,)1 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 (9). It now follows from (8) and (9) that n k k k f(j) χ(m) χ(m) j1 l0 m n/ l (m,)1 l0 m A/ l +B because χ() 0 and a+ ma+1 χ(m) 0 for any a. Now since so we have a l A/ l (a 0 + a a l l )/ l < a l + 1 n f(j) j1 This roves the theorem. l0 χ(m) l0 m A/ l k 1 χ(m) N(n, l) χ(m). m a l m l l0 In this language, Theorem 7 can be stated as follows. Corollary 3. If N(n, l) is the number of digits l in the base exansion of n, then n 1 (10) L (n) λ (j) N(n, l) m. j1 l0 m l As an alication of this theorem consider 3. Alication 1. The value of L 3 (n) is equal to the number of 1 s 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 (6) (or equivalently, the right-hand side of (10)) is equal to N(n, 1). The result then follows from Theorem 7 (or equivalently Corollary 3). 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 negative more often than not. Indeed, we may classify all for which L (n) 0 for all n N. Theorem 9. The function L (n) 0 for all n exactly for those odd rimes for which 1 2 k for all 1 k.

10 10 PETER BORWEIN, STEPHEN K.K. CHOI, AND MICHAEL COONS Proof. We first observe from (5) that if 0 r <, then r r l λ (l). From theorem 7, n λ (l) a 0 a 0 a 1 λ (l) + λ (l) l a 1 + l a k a k because all a j are between 0 and 1. The result then follows. λ (l) l Corollary 4. For n N, we have 0 L 3 (n) [log 3 n] + 1. Proof. This follows from Theorem 9, Alication 1, 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 5. 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 2. The reason for secific values in the roceeding 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 3. 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. Theorem 10. If L +, then 3 (mod 4). Proof. Note that if 1 (mod 4), then for all 1 a 1, so that 2 1 a 0.

11 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1} 11 Consider the case that so that ( 1)/2 1. Then 2 1 On the other hand, if have so that ( ( 1)/ ( 1)/2 1 < 0. ) 1, then since ( 1)/ , ( 1)/ ( 1)/2+1, we + 1, 1 < A bound for L (n) Above we were able to give exact bounds on the function L (n). As exlained in Remark 2, this is not always ossible, though an asymtotic bound is easily attained with a few reliminary results. Lemma 3. For all r, n N we have L ( r n) L (n). Proof. For i 1,..., 1 and k N, λ (k+i) λ (i). This relation immediately gives for k N that L ((k + 1) 1) L (k) 0, since L ( 1) 0. Thus r n L ( r n) λ (k) k1 r 1 n k1 The lemma follows immediately. λ (k) r 1 n k1 λ ()λ (k) r 1 n k1 λ (k) L ( r 1 n). Theorem 11. The maximum value of L (n) for n < i occurs at n k σ( i 1 ) with value max n< i L (n) i max n< L (n), where σ(n) is the sum of the divisors of n. Proof. This follows directly from Lemma 3. Corollary 6. If is an odd rime, then L (n) log n; furthermore, max L (n) log x.

12 12 PETER BORWEIN, STEPHEN K.K. CHOI, AND MICHAEL COONS 6. Concluding remarks Throughout this aer, we were interested in estimates concerning the artial sums of multilicative functions in F({ 1, 1}). An imortant question is: what can be said about the growth of f(n) for any function f F({ 1, 1})? This question goes back to Erdős. Indeed, Erdős [5] states: Finally, I would like to mention an old conjecture of mine: let f(n) ±1 be an arbitrary number theoretic function [not necessarily multilicative]. Is it true that to every c there is a d and an m so that m f(kd) > c? k1 I have made no rogress with this conjecture. Concerning this conjecture, he adds in [6] that the best we could hoe for is that m max f(kd) > c log n. md n k1 We remark that these questions can also be asked for functions f(n) which take kth roots of unity as values rather than just ±1. However, very little is yet known for this case. If we restrict Erdős question to the class of comletely multilicative functions, it may be ossible to rovide an answer. Which would be interesting. Recall that for an odd rime and λ a character like function, we have max N x λ (n) log x, n N so that the class of character like functions satisfies Erdős conjecture. These functions are art of a larger class of functions called automatic functions (see [1]). Let T (t(n)) n 1 be a sequence with values from a finite set. Define the k kernel of T as the set T (k) {(t(k l n + r)) n 0 : l 0 and 0 r < k l }. Given k 2, we say a sequence T is k automatic if and only if the k kernel of T is finite. As a generalization of our result concerning character like functions, we ask the following question. Question 3. Let f F({ 1, 1}) be k automatic for some k 2, and suose that f(n) o(x). Then is it true that max N x n N f(n) log x? References 1. J.-P. Allouche and J. Shallit. Automatic sequences. Cambridge University Press, Cambridge, Paul T. Bateman and Harold G. Diamond, Analytic number theory, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004, An introductory course. 3. P. Borwein, S. Choi, B. Rooney, and A. Weirathmuller, The Riemann Hyothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, vol. 27, Sringer, New York, P. Borwein, R. Ferguson, and M.J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Com. (2007), to aear.

13 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN { 1, 1} P. Erdős. Problems and results on combinatorial number theory. In A survey of combinatorial theory (Proc. Internat. Symos., Colorado State Univ., Fort Collins, Colo., 1971), ages North-Holland, Amsterdam, P. Erdős and R. L. Graham. Old and new roblems and results in combinatorial number theory, volume 28 of Monograhies de L Enseignement Mathématique [Monograhs of L Enseignement Mathématique]. Université de Genève L Enseignement Mathématique, Geneva, Andrew Granville and K. Soundararajan, Motivating the multilicative sectrum, Toics in number theory (University Park, PA, 1997), Math. Al., vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, Andrew Granville and K. Soundararajan, The sectrum of multilicative functions, Ann. of Math. (2) 153 (2001), no. 2, Andrew Granville and K. Soundararajan, Decay of mean values of multilicative functions, Canad. J. Math. 55 (2003), no. 6, G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth ed., The Clarendon Press Oxford University Press, New York, C. B. Haselgrove, A disroof of a conjecture of Pólya, Mathematika 5 (1958), A. E. Ingham, On the difference between consecutive rimes., Quart. J. Math. Oxford 8 (1937), Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, Edmund Landau, Neuer Beweis der Gleichung P µ(k) 0, Inaugural-Dissertation, Berlin, k Edmund Landau, Über die Äquivalenz zweier Hautsätze der analytischen Zahlentheorie, Wien. Sitz. 120 (1911), Nathan Ng, The distribution factor of values of the summatory function of the Möbius function, Notes of the Canad. Math. Soc. 34 (2002), no. 5, Nathan Ng, The distribution of the summatory function of the Möbius function, Proc. London Math. Soc. (3) 89 (2004), no. 2, Hans Carl Friedrich von Mangoldt, Beweis der Gleichung P µ(k) k0 0, Proc. Royal Pruss. k Acad. of Sci. of Berlin (1897), Aurel Wintner, The Theory of Measure in Arithmetical Semi-Grous, ublisher unknown, Baltimore, Md., E. Wirsing, Das asymtotische Verhalten von Summen über multilikative Funktionen. II, Acta Math. Acad. Sci. Hungar. 18 (1967), Deartment of Mathematics, Simon Fraser University, B.C., Canada V5A 1S6 address: borwein@cecm.sfu.ca, kkchoi@math.sfu.ca, mcoons@sfu.ca