Fundamental properties of Björner's complexes

Topology and its Applications 272 (2020) 107055

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Topology and its Applications www.elsevier.com/locate/topol

Fundamental properties of Björner’s complexes Kazunori Noguchi Waseda University, Tokyo, Japan

a r t i c l e

i n f o

Article history: Received 23 February 2019 Received in revised form 5 August 2019 Accepted 3 January 2020 Available online 9 January 2020 MSC: 55U05

a b s t r a c t We study fundamental properties of Björner’s complexes {Δn }n≥1 . These simplicial complexes encode significant. The Prime Number Theorem and the Riemann Hypothesis are equivalent to certain estimates of the reduced Euler characteristics of these complexes as n → ∞. In this paper, we show two facts: the dimension of Δn is approximated by log n/ log log n, and that the number of the maximal dimensional simplices in Δn is less than some constant to the dimension of Δn . © 2020 Elsevier B.V. All rights reserved.

Keywords: Björner’s complexes Prime numbers

1. Introduction Topology can be found in number theory. For example, one can show that there are infinitely many primes using topology [4]. The following is a sketch of the proof. 1. Give a certain topology, not discrete, using arithmetic progressions to Z. 2. Then any nonempty open set in the space is an infinite set. 3. If we assume that the set of primes is finite, then the set {1, −1} is open. This contradicts the second step. Even though we do not really need topology to prove this fact, the proof is interesting. The next examples are Björner’s complexes [2]. For a squarefree positive integer k, let P (k) be the set of prime factors of k. For any n ≥ 1, define an abstract simplicial complex Δn to be the set of P (k) for all squarefree integers 1 ≤ k ≤ n. Let X be a finite set. An abstract simplicial complex Δ is a family of subsets of X that is closed under taking subsets. Namely, if σ belongs to Δ and τ is a subset of σ, then τ also does. We are allowed to take an empty set as τ . If the cardinality of σ is i + 1, then we call σ an i-simplex. E-mail address: [email protected]. https://doi.org/10.1016/j.topol.2020.107055 0166-8641/© 2020 Elsevier B.V. All rights reserved.

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K. Noguchi / Topology and its Applications 272 (2020) 107055

The dimension of Δ is the maximum integer d such that there exists a d-simplex of Δ. Note that one is squarefree; therefore, all Δn contain P (1) = ∅. For example, Δ6 = {∅, {2}, {3}, {5}, {2, 3}} , and it is visualized as follows. 2

3

5

Furthermore, Δ15 is 7

3

2

5

11

13,

and we can find the cycle by {2, 3}, {2, 5}, and {3, 5}. It is very important if a space has cycles or not in topology. However, if n = 30, then the cycle is filled by the triangle {2, 3, 5}. As n grows, cycles are born and filled. The behavior is very complicated. Björner gave the following equivalences: Theorem 1.1 (Björner, 2011; see [2]). Prime Number Theorem ⇔ χ (Δn ) = o(n) 1

Riemann Hypothesis ⇔ χ (Δn ) = O(n 2 +ε ). The (reduced) Euler characteristic χ (Δ) of an abstract simplicial complex Δ is given by χ (Δ) = #σ−1 (−1) . σ∈Δ In [7], topological formulations of famous conjectures in number theory, for example, the Goldbach conjecture, are given by Pakianathan and Winfree. Moreover, Ehrenborg, Govindaiah, Park, and Readdy studied the van der Waerden complexes, and this work also belongs to the intersection of topology and number theory [3]. In this paper, we investigate fundamental properties of Björner’s complexes Δn . We first consider the growth of the dimension of Δn as n → ∞. From the basic definitions, we have 

dim Δn = 0 ⇔ 2 ≤ n ≤ 5 dim Δn = 1 ⇔ 6 ≤ n ≤ 29 dim Δn = 2 ⇔ 30 ≤ n ≤ 209 dim Δn = 3 ⇔ 210 ≤ n ≤ 2309 dim Δn = 4 ⇔ 2310 ≤ n ≤ 30029 .. . We can easily see that the dimension of Δn diverges to ∞ as n → ∞. However, how fast does the sequence {dim Δn }n≥1 diverge? The following is the answer.

K. Noguchi / Topology and its Applications 272 (2020) 107055

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Theorem 1.2. We have log n +O dn = dim Δn = log log n



log n (log log n)2

 .

Hence the dimension of Δn is almost 

log n log log n



if n is large. This result is similar to Theorem 3.10 and 3.11 of [3]. The proof of Theorem 1.2 is elementary, and the facts that are used in the proof are also elementary. Next we consider the number of dn -simplices of Δn . More generally, let Δ be a simplicial complex of dimension d. We denote the number of i-simplices of Δ by fiΔ . The number of d-simplices in Δ is more important than that of i-simplices for any 0 ≤ i ≤ d −1 when we study barycentric subdivision. For example, put d = 2 and let Δ(k) be the k-times subdivided complex of Δ. Then we have the following recurrences: (k+1)

=6f2Δ

(k+1)

=6f2Δ

(k+1)

=f2Δ

f2Δ f1Δ f0Δ

(k)

(k)

(k)

(k)

+ 2f1Δ (k)

+ f1Δ

(k)

+ f0Δ

for k ≥ 0. Hence we obtain (k)

f2Δ

(k) f1Δ

(k)

f0Δ

=f2Δ 6k

 3 Δ k = f2 6 + f1Δ − 2  1 = f2Δ 6k + f1Δ − 2

 3 Δ k f 2 2 2  3 Δ k  Δ f2 2 + f0 − f1Δ + f2Δ . 2

When k is large, 6k is much larger than 2k ; therefore, the number of i-simplices of Δ(k) is almost determined by f2Δ and 6k . In this case, f1Δ and f0Δ have very few influence on the result. Let us consider the number theoretic meaning of fdΔnn ; it is the number of squarefree positive integers of d weight dn + 1 less than or equal to n. If n = pe11 pe22 . . . pedd , then the weight of n is defined by i=1 ei . For example, fdΔ66 = f1Δ6 = 1 and fdΔ1515 = f1Δ15 = 4. The numbers fdΔnn can decrease as n grows. For example, fdΔ2929 = f1Δ29 = 7, however fdΔ3030 = f2Δ30 = 1. We prove the following result. Theorem 1.3. We have  fdΔnn = O C dn for some constant C > 0. The following result follows from Theorem 1.2 and 1.3. Corollary 1.4. We have   log n = O exp A log log n 

fdΔnn

K. Noguchi / Topology and its Applications 272 (2020) 107055

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for some constant A > 0. Since n = exp(log n), we can see fdΔnn is smaller than n. The proof is also elementary. Finally, [1,6,8] are basic to read this paper. 2. Proof of Theorem 1.2 We prove Theorem 1.2. Proof of Theorem 1.2. By Theorem 4.7 of [1], we have C1 n log n < pn < C2 n log n for some constants C1 , C2 , and any n ≥ 2. We can take C1 = Hence, we have C1d+1 (d + 1)!

d+1

1 6

and C2 = 24. Then, C1 < p1 = 2 < C2 .

log m < n < C2d+2 (d + 2)!

m=2

Since the function

log x log log x

d+2

log m.

(1)

m=2

is monotonically increasing in [ee , ∞), we have

d+2 (d + 2) log C2 + log(d + 2)! + m=2 log log m log n  < d+2 log log n log (d + 2) log C2 + log(d + 2)! + m=2 log log m if n ≥ ee . By integral test, we have d+1

d log log d − Li d + C ≤

log log m ≤ (d + 1) log log(d + 1) − Li(d + 1) + C,

m=2

where Li x =

x

dt 2 log t

and C =

e

dt 2 log t

+ log log 2. By Stirling’s formula, we have, for ε > 0,

(d + 2) log C2 + (d + 3) log(d + 3) + (d + 2) log log(d + 2) log n < log log n log(d + 3) + log log(d + 3) <(1 + ε)d

(2)

if n is sufficiently large. If we replace ε by 2 log C2 , log d + log log d the inequality (2) holds for all large enough n, d. Hence, we have log n 1 1 + ε log log n log n ε log n = − log log n 1 + ε log log n log n 2 log C2 log n > − . log log n log d log log n

d>

K. Noguchi / Topology and its Applications 272 (2020) 107055

If we put ε =

1 2

5

and take logarithm in (2), we have  log d > log

2 log n 3 log log n

 >

1 log log n. 2

(3)

Hence, we obtain d>

log n log n − 4 log C2 . log log n (log log n)2

By the left inequality of (1), we can similarly obtain log n ≥ (1 − ε)d, log log n and we can replace ε by 2 log C1 . log d + log log d By (3), we obtain d≤

log n log n + 4 log C1 . log log n (log log n)2

Hence, the result follows. 2 3. Proof of Theorem 1.3 We prove Theorem 1.3. Denote πd (x) the number of positive squarefree integers of weight d not exceeding a real number x. Lemma 3.1. We have  πd (p1 p2 . . . pd+1 ) = O C d for some constant C > 0. Proof. Suppose that a sequence of primes q1 , q2 , . . . , qd satisfies q1 q2 . . . qd ≤ p1 p2 . . . pd+1 and q1 < · · · < qd . We count the number of such (q1 , . . . , qd ). The greatest member qd does not exceed pd3 , since q1 q2 . . . qd ≥ p1 p2 . . . pd−1 qd ; that is, qd ≤ pd pd+1 ≤ C22 (d + 1)2 log2 (d + 1). Hence, we choose q1 , q2 , . . . , qd from the set {p1 , p2 , . . . , pd3 }. 2 We choose m satisfying m ≥ eCC1 2 . The number of i, 1 ≤ i ≤ d, such that qi ≥ pmd is smaller than [log d]. Indeed, if q1 , q2 , . . . , qd−[log d] < pmd and qd−[log d]+1 , . . . , qd ≥ pmd , then we have q1 q2 . . . qd ≥ p1 p2 . . . pd−[log d] pmd pmd+1 . . . pmd+[log d] . Hence, it must be that

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K. Noguchi / Topology and its Applications 272 (2020) 107055

pmd . . . pmd+[log d] ≤ pd−[log d]+1 . . . pd+1 . The left hand side is, at least, pmd . . . pmd+[log d] ≥ (C1 md log d)

[log d]

,

and the right hand side is, at most, [log d]+1

pd−[log d]+1 . . . pd+1 ≤ (C2 d log d) However, the condition m ≥

e2 C 2 C1

.

implies pmd . . . pmd+[log d] > pd−[log d]+1 . . . pd+1 .

Hence, the claim follows. By Stirling’s formula, we have   3[log d] md d πd (p1 . . . pd+1 )  d [log d]! md dd 3 log d d d! ed md dd 3 log d  d dd



C d for some constant C > 0. Hence, the result follows. 2 Proof of Theorem 1.3. By the definition of Δn , we have fdΔnn = πdn +1 (n) ≤ πdn +1 (p1 p2 · · · pdn +2 ). The result follows from Lemma 3.1. 2 Remark 3.2. The following approximation is well-known: πd (x) ∼

x (log log x)d−1 . log x (d − 1)!

See Theorem 437 of [5]. Hence πd (x) is approximated by the right-hand side if x is sufficiently large, however, x = p1 p2 · · · pd+1 is not large enough. Indeed, by Theorem 4.7 of [1] and Stirling’s formula, we have x (log log x)d−1 ≥ exp (Bd log log d) log x (d − 1)! for some constant B > 0, and the right-hand side is not better than Lemma 3.1. Acknowledgement I wish to thank the referees to give me helpful comments. I am a researcher at the Research Institute of Kinesiology. I would like to thank Hideo Takaoka for all his help and instruction in the YURU PRACTICE. The YURU PRACTICE has always supported me.

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References [1] T.M. Apostol, Introduction to Analytic Number Theory, Springer, 1976. [2] A. Björner, A cell complex in number theory, Adv. Appl. Math. 46 (2011) 71–85. [3] R. Ehrenborg, L. Govindaiah, Peter S. Park, M. Readdy, The van der Waerden complex, J. Number Theory 172 (2017) 287–300. [4] H. Fürstenberg, On the infinitude of primes, Am. Math. Mon. 62 (1955) 353. [5] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford University Press, Oxford, 2008, Revised by D. R. Heath-Brown and J. H. Silverman, with a foreword by Andrew Wiles. [6] D. Kozlov, Combinatorial Algebraic Topology, Springer-Verlag, 2008. [7] J. Pakianathan, T. Winfree, Threshold complexes and connections to number theory, Turk. J. Math. 37 (3) (2013) 511–539. [8] E. Spanier, Algebraic Topology, McGraw-Hill, 1966.