Advances in Applied Mathematics

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1 Advances in Applied Mathematics Contents lists available at ScienceDirect Advances in Applied Mathematics wwwelseviercom/locate/yaama A cell complex in number theory Anders Björner a,b, a Institut Mittag-Leffler, Auravägen 17, S Djursholm, Sweden b Kungl Tenisa Högsolan, Matematisa Inst, S Stocholm, Sweden article info abstract Article history: Available online 20 October 2010 Dedicated to Dennis Stanton on the occasion of his 60th birthday MSC: 05E99 11A99 Keywords: Mertens function Liouville function Multicomplex Cellular realization Let n be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility It is nown that the asymptotic rate of growth of its Euler characteristic the Mertens function is closely related to deep properties of the prime number system In this paper we study the asymptotic behavior of the individual Betti numbers β n and of their sum We show that n has the homotopy type of a wedge of spheres, and that as n β n = 2n π 2 + O n θ, for all θ> Furthermore, for fixed, β n n log log n 2logn! As a number-theoretic byproduct we obtain inequalities σ +1 n σ n/2, where σ n denotes the number of squarefree integers n with prime factors, and is a certain combinatorial shadow function We also study a CW complex n that extends the previous simplicial complex In n all numbers n correspond to cells and its Euler characteristic is the summatory Liouville function This Research supported by the Knut and Alice Wallenberg Foundation, grant KAW * Address for correspondence: Kungl Tenisa Högsolan, Matematisa Inst, S Stocholm, Sweden addresses: bjorner@maththse, bjorner@mittag-lefflerse /$ see front matter 2010 Elsevier Inc All rights reserved doi:101016/jaam

2 72 A Björner / Advances in Applied Mathematics cell complex n is shown to be homotopy equivalent to a wedge of spheres, and as n β n = n 3 + O n θ, for all θ> Elsevier Inc All rights reserved 1 Introduction Let Mn def = n =1 μ, where μ is the number-theoretic Möbius function The rate of growth of the function Mn is of great interest and importance in number theory, as is clear from the following facts: Prime Number Theorem Mn εn, for all ε > 0 and all sufficiently large n Riemann Hypothesis Mn n 1/2+ε, for all ε > 0 and all sufficiently large n The ey questions concerning the growth of Mn are subtle For instance, Mertens conjectured in 1897 that Mn n 1/2 for all sufficiently large n This conjecture was disproved in 1985 by Odlyzo and te Riele See eg the boos by Hardy and Wright [5] and Ivić [6, especially 19] for more information about these matters This paper has its genesis in the observation that the Mertens function Mn can be interpreted as the Euler characteristic of a simplicial complex Namely, for each positive squarefree integer, let P be the set of its prime factors For instance, P165 ={3, 5, 11} Then the set family n def = { P: is squarefree and n } is closed under taing subsets In other words, it is an abstract simplicial complex Remar: The integer 1 is squarefree, so we include P1 = in n Since, by definition, it follows that { μ = 1 P, if is squarefree, 0, otherwise Mn = χ n, 11 where χ n is the reduced Euler characteristic the ordinary Euler characteristic minus one Let β n denote the -th Betti numbers of reduced simplicial homology, ie, β n def = ran H n, Z Then, from the Euler Poincaré formula and Eq 11 we have Mn = β n 12 Thus, important number-theoretic propositions, such as the Prime Number Theorem and the Riemann Hypothesis, are equivalent to statements about the asymptotic rate of growth of the Euler characteristic χ n It therefore seems reasonable to inquire about the Betti numbers of the complex n and their asymptotics

3 A Björner / Advances in Applied Mathematics We prove the following estimates: As n 0 even β n = 2n π 2 + O n θ, for all θ> 17 54, β n n π 2, β n n π 2, for fixed : β n n 2logn log logn! Unfortunately these results fall short of shedding new light on the rate of growth of the Euler characteristic Mn Perhaps a study of deeper topological invariants of n could add something of value Let σ n be the number of squarefree integers n with prime factors It turns out that the homology information of n can be expressed in terms of these numbers, which, in turn, leads to some purely number-theoretic consequences Namely, for certain shadow functions and,well nown in extremal combinatorics and defined in Section 4, we obtain inequalities 1 σ +1 2 σ 2+2 n σ n/2, n + σ 2+1 n σ 2 n/2 + σ2 1 n/2 In Section 5 we study a CW complex n that extends the previous simplicial complex In n all numbers n correspond to cells and its Euler characteristic is the summatory Liouville function We get a sequence of embeddings 2 3 n that can be seen as a filtration of the join of denumerably many copies of certain infinite-dimensional spaces such as S or RP The cell complex n is shown to be homotopy equivalent to a wedge of spheres, and as n β n = n 3 + O n θ, for all θ> The construction of n is based on a more general construction for multicomplexes given in [3] In the last section we recall and expand on the details of the construction from [3] Helpful remars from anonymous referees are gratefully acnowledged 2 Preliminaries Later on we need to refer to a few results from combinatorial topology and number theory Here this information is recalled A simplicial complex on a linearly ordered vertex set x 1, x 2,,x t is said to be shifted if for every F : i < j, x i / F and x j F F \{x j } {x i }

4 74 A Björner / Advances in Applied Mathematics Theorem 21 [2, p 292] Suppose that is a shifted complex on the vertex set x 1, x 2,,x t Then has the homotopy type of a wedge of spheres Its Betti numbers are β = number of F such that F {x 1 } / and F = + 1 The number Ωn of prime factors of a positive integer n is called its weight Ie, by definition, Ω p e 1 i 1 p e 2 i 2 p e i = e1 + e 2 + +e It is convenient to define the following integer-valued functions for all positive real numbers x not only for integers Definition 1 For x R +,let 1 σ x def = the number of squarefree integers in 0, x], 2 σ x def = the number of squarefree integers in 0, x], 3 σ x def = the number of squarefree integers in 0, x] of weight, 4 σ x def = the number of squarefree integers in 0, x] of weight, 5 and analogously σ even x and σ even x Note that σ x = 0forx < 1 and similarly for the other functions The following estimates belong to the classics of number theory Theorem 22 As x we have that a Gegenbauer, 1885; Jia, 1993; see [5, pp 355, 359] b Landau, 1900; see [5, p 491] For fixed, σ x = 6x π 2 + O x θ, for all θ> 17 54, σ x x log log x 1 log x 1! Remar 23 i Gegenbauer s estimate of the error term was O n The sharper exponent cited here is due to Jia ii Landau s asymptotic formula for σ x was conjectured by Gauss Note that the = 1caseis the Prime Number Theorem Estimates of the error term exist but will not be used in this paper 3 The number-theoretic simplicial complex n Some of the basic concepts of elementary number theory have direct geometric meaning for the complex n In fact, one can read parts of the boo by Hardy and Wright [5] as describing the size and shape of this complex For instance, the dimension of a simplex P is the weight of minus one: dim P = Ω 1

5 A Björner / Advances in Applied Mathematics and the typical dimension of a simplex in n is circa log logn Forintegers 1, let #! denote the primorial number p 1 p 2 p, that is, the product of the first prime numbers Then dim n = max{: #! n} 1 The round numbers [5, p 476] less than n roughly correspond to the high-dimensional simplices in n For example, dim 10 7 = 7 and a typical simplex of 10 7 is 2-dimensional, whereas dim = 44 and a typical simplex is of dimension 4 or 5 Compare this to the discussion in [5, p 477], where in passing it is mentioned that is roughly the number of protons in the universe Theorem 31 The complex n is shifted Its Betti numbers are β n = σ +1 n σ+1 n/2 31 Proof The vertices of n are the prime numbers p 1 = 2, p 2 = 3, p 3 = 5,Ifforasquarefreenumber n one of its prime factors is replaced by a smaller prime number, then the new number so obtained satisfies <, and in particular < n Hence n is shifted According to Theorem 21 the Betti numbers are β n = # { squarefree integers b n such that 2b n and Ωb = + 1 }, which agrees with the stated expression The plan from here on is to get rid of the condition in formula 31 We see to instead express the Betti numbers in terms of the standard number theoretic functions σ x and σ x, so that we can benefit from the nown estimates for these functions Lemma 32 Proof Multiplication by 2 gives a bijective map σ even x = σ 1 x/2 { squarefree numbers in 0, x/2] of weight 1 } { even squarefree numbers in 0, x] of weight } Lemma 33 Proof Lemma 32 implies that σ x = σ x σ 1 x/2 + σ 2 x/4 σ 3 x/8 +, 32 σ x = σ x σ x/2 + σ x/4 σ x/ σ x = σ x + σ 1 x/2, from which Eq 32 follows Eq 33 is then obtained by summation over all

6 76 A Björner / Advances in Applied Mathematics Theorem 34 As n we have that 0 β n = 2n π 2 + O n θ, for all θ> Proof Let N def = log 2 n We then have that n 2 N < 2n Using Theorem 22 and Lemma 33 we compute Hence, by Eq 31 σ n = = N n 1 i σ i=0 2 i N [ θ ] 6 1 i n n + π 2 2 i O 2 i i=0 = 6n π 2 N 1 i N θ n + O 2 i=0 = 6n 2 π i=0 2 i 1 N+1 N + n θ O 2 = 4n 1 π + O1 + 2 nθ O 1 2 θ = 4n π 2 + O n θ N iθ i=0 0 β n = σ n σ n/2 = 4n π 2 + O n θ 2n π 2 O n/2 θ Theorem 35 As n we have that even β n n π 2 and β n n π 2 Proof Let an denote the first sum and bn the second Theorem 34 and the Prime Number Theorem show, respectively, that an + bn n 2 π 2 and an bn n = Mn n 0 as n Hence, 2an an + bn an bn = + n n n 2 π and similarly for bn

7 A Björner / Advances in Applied Mathematics Theorem 36 For fixed and n, β n n log logn 2logn! Proof We begin with a small auxiliary computation Using Theorem 22 we have that, as n, σ i n/2 j σ n n/2 j logn/2 j n logn log logn/2 j i 1 i 1! log log n 1 1! Hence, for fixed i, j 0 = 1 log n log logn/2 j i 1 2 j logn/2 j 1 2 i log logn log logn i 1 { 1, if i = 0, 2 j 1 1 0, if i > 0 σ i n/2 j { 1/2 lim = j, if i = 0, n σ n 0, if i > 0 34 Eqs 31 and 32 imply β 1 n = σ = n σ n/2 1 j[ σ j n/2 j σ ] j n/2 j+1 j=0 Using 34 we have that β 1 n σ n = 1 σ n/2 σ n + [ σ 1 j j n/2 j σ n j=1 σ jn/2 j+1 ] σ n = 1 2 Hence, β 1 n 1 σ n n log log n 1 2 2logn 1! 4 Some number-theoretic consequences The fact that the Betti numbers of the complex n are determined by counting certain squarefree integers has some purely number-theoretic implications Namely, nowing the number of such integers of weight + 1 in the interval 0, n] one obtains a lower bound for the number of such integers of weight in the interval 0,n/2]

8 78 A Björner / Advances in Applied Mathematics We need to recall the following definitions Two number-theoretic functions n and n are defined in the following way For n, 1 the integer n caninauniquewaybeexpressedinthe following form n = a + where a > a 1 > > a i i 1 Then let: a 1 1 n def a = a 1 2 ai i, ai + +, i 1 and 1 n def = a a ai 1 i 1 Also, we let 1 0 = 1 0 = 0 Theorem 41 For all 1 we have that 1 σ +1 2 σ 2+2 n σ n/2, n + σ 2+1 n σ n/2 + σ n/2 Proof For an arbitrary finite simplicial complex with β 0,β 1,,letfor 0 f -vector f 0, f 1, and Betti numbers χ 1 def = j 1 j f j β j The following relations appear as Theorem 11 in [2] and Theorem 33 in [3], respectively For all 1, χ + β χ 1, 41 f β 2 f 2 1 β Let us now see what these relations mean for the particular complex n : χ 1 = j 1 j σ j+1 n β j n = j 1 j σ even = 1 j σ j n/2 + σ j+1 n/2 = σ n/2 j j+1 n + σ j+1 n/2 Hence, χ + β = σ +1 n/2 + σ +1 n σ+1 n/2 = σ +1 n, σ 2+2 n + β 2 n = σ 2+2 n + σ 2+1 n σ2+1 n/2 = σ 2+2 n + σ2+1 n, σ 2 n β 2 1 n = σ 2 n σ 2 n σ 2 n/2 = σ 2 1 n/2 + σ2 n/2 Inserting these evaluations into relations 41 and 42 one obtains the theorem

9 A Björner / Advances in Applied Mathematics The number-theoretic cell complex n The system of squarefree numbers less than or equal to n and ordered by divisibility presents itself immediately as a simplicial complex The same is not true for the larger system of all such numbers However, a construction is nown [3] which produces a CW complex of a similar nature The CW structure is no longer uniquely defined However, if one demands that its closed i-dimensional cells are i 1-connected for all i then the complex is uniquely determined up to homotopy type The construction from [3] of cellular realizations of multicomplexes is reviewed in Section 6 We tae it here for nown The system of positive integers less than or equal to n, ordered by divisibility, is isomorphic to a multicomplex of monomials Namely, associate an indeterminate x i with each prime number p i n and then extend this to a bijection p e 1 i 1 p e 2 i 2 p e i x e 1 i 1 x e 2 i 2 x e i, i 1 < i 2 < < i Therefore the construction of a cellular realization Γ, reviewed in Section 6, is applicable Definition 2 Let n denote the CW complex ΓM, where M is the multicomplex of positive integers less than or equal to n, and the construction of Γ is based on the choice of a well-connected CW string Here is a summary of the main features of n, see Section 6 for further details 1 The positive integers n are in bijection with the closed cells c of n 2 dim c = Ω 1 3 The integer 1 divides 2 if and only if c 1 c 2 4 The cell c has the following homotopy type: { S c d 1, if is a full square and Ω = d, apoint, otherwise 5 In case we are using the S string even more is true If Ω = d, then the cell c has the following homeomorphy type sphere or ball: 6 The Euler characteristic is { c S = d 1, B d 1, if is a full square, otherwise χ n = # { n, Ω is } # { n, Ω is even } The Euler characteristic of n is equivalent to one of the classic functions of number theory The summatory Liouville function Ln is defined as follows: Ln = n 1 Ω See [4] for information about this function So, χ n = Ln Thus, the summatory Liouville function Ln plays for the cell complex n thesamerolethatthe Mertens function Mn plays for the simplicial complex n The two functions are related as follows

10 80 A Björner / Advances in Applied Mathematics Proposition 51 n n Ln = M r 2 n n and Mn = μrl r 2 Proof Let [n] def def ={1, 2,,n} and [n] sqf ={ [n] is squarefree} Every positive integer can be uniquely factored as a product of a full square r 2 and a squarefree integer s: = r 2 sthisimpliesthe following set-theoretic disjoint union decomposition n [n]= [ ] n r 2 r 2 sqf Since Ω = Ωr 2 s Ωs mod 2, the first formula follows Then the second one is obtained by Möbius inversion of the form presented in [5, p 307] The rates of growth of the two functions Ln and Mn are essentially identical, as the following proposition shows Proposition 52 For every ε > 0,asn, Mn = o n 1/2+ε if and only if Ln = o n 1/2+ε Proof In what follows all numbers that are not integers are to be rounded down to the closest smaller integer So, n is to be read n, etc Let ε > 0 and assume that Mn = on 1/2+ε as n Choose any δ>0, and let N be such that 1 r=n+1 r 1+2ε <δ and 1/N δ Then, if n is large enough that n > N and we have that M n r 2 n r 2 1/2+ε 1/N2, for all r {1, 2,,N}, n Ln 1 M n r 2 n1/2+ε r 1+2ε n 1/2+ε r 2 N N 2δ 1 M n r 2 r 1+2ε n 1/2+ε + r 2 1 N 2 + δ 1 r 1+2ε r=n+1 The same computation, exchanging the roles of Mn and Ln, gives by Proposition 51 the opposite implication By way of the ε = 1/2 case of Proposition 52, the Prime Number Theorem implies that Ln = on, as n 51

11 A Björner / Advances in Applied Mathematics Also, one can conclude from Proposition 52 that the Riemann Hypothesis is equivalent to Ln = O n 1/2+ε, as n, for every ε > 0 52 The preceding shows that the Euler characteristic of the cell complex n has the same numbertheoretic relevance as that of the simplicial complex n This motivates seeing information about the Betti numbers of n, as we did in Section 3 for n Theorem 53 We have the following homotopy equivalence n susp 2Ωr n/r 2 r 2 n Proof This is a direct consequence of Theorem 64 Corollary 54 n has the homotopy type of a wedge of spheres, and n β n = β 2Ωr n/r2 Proof Combine with Theorem 53 the information coming from Theorems 31 and 65 Theorem 55 As n we have that 0 β n = n 3 + O n θ, for all θ> Proof Let ψ> 17 Using Theorem 34 and Corollary 54 we get: 54 β n = 0 n β n/r 2 n 2n/r 2 ψ n = + O π 2 = 2n n 1 π 2 r + 2 n O n ψ = 2n π 2 π O 1 n + r 2 O n 1/2+ψ r= n r 2 = n 3 + O n 1/2 + O n 1/2+ψ = n 3 + O n θ, where θ = ψ> = 22 27

12 82 A Björner / Advances in Applied Mathematics Theorem 56 As n we have that β n n 6 and β n n 6 even Proof Let an denote the first sum and bn the second Theorem 55 and Eq 51 show, respectively, that an + bn 1 n 3 and an bn n = Ln n 0 as n Hence, the proof can be completed as that of Theorem 35 6 Cellular realization of multicomplexes In this section we review the construction of CW complexes from [3] We expand on some details and present a new result concerning their homotopy type By a CW string we shall mean an infinite sequence of embeddings such that c 0 c 1 c j 1 {c j } j 0 are the closed cells of a CW decomposition of the colimit c j 2 dim c j = j, forall j 0 The CW string is said to be well connected if 3 c j is j 1-connected, for all j 0 Lemma 61 A CW string is well connected if and only if c j is contractible for all even j and has the homotopy type of the j-sphere for all j Proof It is nown that a j 1-connected and j-dimensional space has the homotopy type of a wedge of j-dimensional spheres The number of spheres is given by the Euler characteristic Since the string has exactly one cell in each dimension, the reduced Euler characteristic of c j is0foralleven j and 1 for all j Here are three constructions of CW strings The first one was suggested to us by Michael J Fal private communication The other two appear in [3] 1 The S string This is a CW decomposition of the infinite-dimensional sphere S = {x 1, x 2, x n = 0forsufficientlylargen, and x x2 2 + =1} The cells are c j = { x 1, x 2, S x = 0 for all > j + 1 }, if j is, c j = { x 1, x 2, S x j+1 0, x = 0 for all > j + 1 }, if j is even It is clear, when j is, how to attach a j + 1-ball onto the sphere c j in order to obtain the ball c j+1 When j is even the attachment map has the following description Note that c j is here the upper hemisphere of S j,the j-dimensional sphere Consider the j + 1-ball B j+1 = { x 1, x 2, S x j+2 0, x = 0 for all > j + 2 },

13 A Björner / Advances in Applied Mathematics ie, the upper hemisphere of S j+1 By switching to polar coordinates for the last two coordinates, the points of S j can be coordinatized S j = { x 1, x 2,,x j 1, r cos θ,r sin θ,0, 0, } S In terms of these coordinates the attachment map α : bdb j+1 = S j c j sends x 1, x 2,,x j 1, r cos θ,r sin θ to x 1, x 2,,x j 1, r cos 2θ,r sin 2θ Thus we have a CW string which is well connected in a strong sense, namely its cells are homeomorphic to balls and spheres of the appropriate dimensions 2 The D string The j-dimensional dunce hat D j is obtained from the j-dimensional simplex a 0, a 1,,a j by identifying all of its j 1-dimensional faces, with their vertices in the induced order Note that D 2 is the usual dunce hat It is shown in [1] that D j is contractible, but not collapsible, for all even j > 0, and that D j is a homotopy sphere for all j Thus, with the obvious attaching maps we have a well-connected CW string D 0 D 1 D j giving a CW decomposition of its colimit D, the infinite-dimensional dunce hat 3 The RP string The standard CW decomposition of infinite-dimensional real projective space, with one cell in each dimension, provides a well-nown example of a CW string RP 0 RP 1 RP j This string is not well connected However, on the level of rational homology H, Q this string behaves in a way that for algebraic purposes parallels that of well-connected strings, see Remar 66 Let M be a multicomplex By this we mean a finite collection of monomials in indeterminates x 1, x 2, closed under divisibility A CW complex ΓM is constructed as follows It depends on a choice of CW string {c j } j 0, which once chosen remains fixed and will not be included in the notation For each indeterminate x i tae a copy {c j x e 1 i 1 x e 2 i 2 x e i M associate the space cm def = c e 1 1 i 1 i } j 0 of the string Then, to each monomial m = c e 2 1 c e 1 i 2 i where denotes the join of topological spaces Note that if m is squarefree then cm is a 1- dimensional simplex, since c 0 is a single vertex for every choice of CW string We say that m is a full square if m = r 2 for some monomial r Lemma 62 [3, Prop 22] The space cm has the following homotopy type: { S cm d 1, if m is a full square of degree d, apoint, otherwise We remar that in case we are using the S string even more is true: cm is homeomorphic to the d 1-sphere or the d 1-ball, respectively The CW complex ΓM associated with M is constructed as follows With each monomial m M associate the space cm, and then glue these together to form ΓM The attaching maps are everywhere the ones coming from how a j-cell is attached to c j 1 i to obtain c j i, and the joins of these maps Proposition 63 See [3, Sect 2] The space ΓM is a CW complex with closed cells cm,m M A cell cm is contained in another cell cm if and only if m divides m

14 84 A Björner / Advances in Applied Mathematics We thin of ΓM as the geometric realization of the multicomplex M Note that if all monomials in M are squarefree then the construction reduces to the usual geometric realization of a simplicial complex Every monomial can be uniquely factorized as a product of a full square monomial and a squarefree monomial: m = r 2 s For each full square monomial r 2 in M we define M r 2 = { s M r 2 s M, s squarefree } Notice that each M r 2 is a simplicial complex, possibly empty Let r denote the degree of a monomial r Theorem 64 Suppose that a well-connected CW string has been used for the construction of ΓMThen ΓM susp 2 r ΓM r 2 r 2 M Proof We are going to use the theory of homotopy colimits of diagrams of spaces The tools from this theory that we need are summarized in an accessible way in [7] We refer the reader to that paper for all explanations of terminology and basic facts used in the sequel We frequently use the fact that the join operation of topological spaces commutes with the needed operations, up to homotopy The multicomplex M, ordered by divisibility is a poset The functor D : m cm and D : m m cm c m givesusadiagramofspacesd : M Top By the Projection Lemma [7, Prop 31] we have that ΓM hocolim D 61 Let E be a diagram over M with constant maps Since dim m <m cm = dim cm 1 and cm is dim cm 1-connected, it follows that any two maps from m <m cm to cm are homotopic In particular, the inclusion map is homotopic to any constant map Therefore, using the Homotopy Lemma [7, Prop 37] and the homotopy extension property, we get hocolim D hocolim E 62 Finally, the Wedge Lemma [7, Lemma 49] implies that hocolim E m M cm M >m, 63 where M >m denotes the order complex of the poset of monomials in M that are strictly above m in the partial order, ie, that are divisible by m Many of the spaces appearing in the wedge are contractible and can therefore be removed without affecting the homotopy type Namely, we have that cm M >m { susp 2 r M >m, if m = r 2 M, apoint, otherwise

15 A Björner / Advances in Applied Mathematics This follows from Lemma 62 and the fact that taing join with the d 1-dimensional sphere is, up to homeomorphism, equivalent to taing d-fold suspension Combining Eqs 61, 62 and 63 we obtain the homotopy equivalence ΓM susp 2 r M >r2 64 r 2 M Thus, the following homotopy equivalences complete the proof: M >r2 Mr 2 ΓM r 2 The first equivalence is implied by [3, Prop 41] The second follows from the fact that the homeomorphism type of a simplicial complex is invariant under barycentric subdivision Corollary 65 [3, p 55] H ΓM, Z = H 2 r ΓMr 2, Z r 2 M Remar 66 The RP string is not well connected, since its cells c j arerealprojectivespaceswith plenty of mod 2 torsion However, the string is rationally well connected, since H c j, Q = { Q, if j is and = j, 0, in all other cases Thus, in terms of rational homology the RP string behaves lie the well-connected strings By replacing the diagrammatic tools used in the proof of Theorem 64 by their homological counterparts one can derive a splitting formula for H Γ M, Q analogous to Corollary 65 The rational Betti numbers produced for the number-theoretic cell complexes using joins of real projective spaces as cells will therefore be the same as those computed in Sections 3 and 5 References [1] RN Andersen, MM Marjanović, RM Shori, Symmetric products and higher-dimensional dunce hats, Topology Proc [2] A Björner, G Kalai, An extended Euler Poincaré theorem, Acta Math [3] A Björner, S Vrećica, On f -vectors and Betti numbers of multicomplexes, Combinatorica [4] P Borwein, R Ferguson, MJ Mossinghoff, Sign changes in sums of the Liouville function, Math Comp [5] GH Hardy, EM Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford Univ Press, Oxford, 2008 [6] A Ivić, The Riemann Zeta-function, Wiley Interscience, New Yor, 1985 [7] V Weler, GM Ziegler, RT Zivaljević, Homotopy colimits comparison lemmas for combinatorial applications, J Reine Angew Math