Number Theory: An Introduction to Mathematics

346 VIII The Geometry of Numbers Proof Letb 1 ,...,bmbe the facet vectors ofΛand put Λ′={x=β 1 b 1 +···+βmbm:β 1 ,...,βm∈Z}. Evi ...

5 Densest Packings 347 G 2 = A−^1 G 1 A. With the aid of results of Minkowski and Jordan it follows that, for a given dimensionn ...

348 VIII The Geometry of Numbers limn→∞γn/n≥ 1 / 2 πe= 0. 0585 ..., and it is possible that actually limn→∞γn/n= 1 / 2 πe. The s ...

5 Densest Packings 349 full-dimensional). We say also thatΛis theorthogonal sumof the latticesΛ 1 andΛ 2. The orthogonal sum of ...

350 VIII The Geometry of Numbers Table 1.Indecomposable root lattices An={x=(ξ 0 ,ξ 1 ,...,ξn)∈Zn+^1 :ξ 0 +ξ 1 +···+ξn= 0 }(n≥ 1 ...

5 Densest Packings 351 Table 2.Densest lattices inRn n Λγn δn 1 A 1 11 2 A 2 ( 4 / 3 )^1 /^2 = 1. 1547 ... 31 /^2 π/ 6 = 0. 9068 ...

352 VIII The Geometry of Numbers 6 Mahler’sCompactnessTheorem............................... It is useful to study not only indi ...

6 Mahler’s Compactness Theorem 353 ofK. We will show that this defines a metric, theHausdorff metric, on the space of all compac ...

354 VIII The Geometry of Numbers The latticeΛkhas at most 2( 2 n− 1 )facet vectors, by Proposition 15. Hence, by restriction to ...

6 Mahler’s Compactness Theorem 355 ‖ykj−xkj‖<δ (j= 1 ,...,m). Sinceykj−xkj∈Λk, this implies that, for all largek,ykj=xkj(j= 1 ...

356 VIII The Geometry of Numbers for all largekwe must have 0 <|det(ak 1 ,...,akn)|< 2 λ(Vk). But ifak 1 ,...,aknwere not ...

7 Further Remarks 357 For otherwise there would exist a sequenceΛkof lattices inF such that either m(Λk)→0ord(Λk)→∞, and clearly ...

358 VIII The Geometry of Numbers Numerous references to the earlier literature are given in Keller [34]. Lagarias [36] gives an ...

7 Further Remarks 359 An interesting subset of all crystallographic groups consists of those generated by reflections in hyperpl ...

360 VIII The Geometry of Numbers packing is a densest packing. A computer-aided proof has recently been announced by Hales [29]. ...

8 Selected References 361 [24] L. Fejes T ́oth,Lagerungen in der Ebene auf der Kugel und im Raum, 2nd ed., Springer-Verlag, Berl ...

362 VIII The Geometry of Numbers [52] A. Schrijver,Theory of linear and integer programming, corrected reprint, Wiley, Chicheste ...

IX The Number of Prime Numbers................................ 1 FindingtheProblem It was already shown in Euclid’sElements(Book ...

364 IX The Number of Prime Numbers and S= 1 + ∑∞ n= 1 1 /(n+ 1 )^2 < 1 + ∑∞ n= 1 ∫n+ 1 n dt/t^2 = 1 + ∫∞ 1 dt/t^2 = 2. (In fa ...

1 Finding the Problem 365 where lognx =(logx)n. By repeatedly integrating by parts it may be seen that, for each positive intege ...