Metamath Proof Explorer


Theorem dchrvmasumlem

Description: The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by n , is bounded. Equation 9.4.16 of Shapiro, p. 379. (Contributed by Mario Carneiro, 12-May-2016)

Ref Expression
Hypotheses rpvmasum.z Z = /N
rpvmasum.l L = ℤRHom Z
rpvmasum.a φ N
dchrmusum.g G = DChr N
dchrmusum.d D = Base G
dchrmusum.1 1 ˙ = 0 G
dchrmusum.b φ X D
dchrmusum.n1 φ X 1 ˙
dchrmusum.f F = a X L a a
dchrmusum.c φ C 0 +∞
dchrmusum.t φ seq 1 + F T
dchrmusum.2 φ y 1 +∞ seq 1 + F y T C y
Assertion dchrvmasumlem φ x + n = 1 x X L n Λ n n 𝑂1

Proof

Step Hyp Ref Expression
1 rpvmasum.z Z = /N
2 rpvmasum.l L = ℤRHom Z
3 rpvmasum.a φ N
4 dchrmusum.g G = DChr N
5 dchrmusum.d D = Base G
6 dchrmusum.1 1 ˙ = 0 G
7 dchrmusum.b φ X D
8 dchrmusum.n1 φ X 1 ˙
9 dchrmusum.f F = a X L a a
10 dchrmusum.c φ C 0 +∞
11 dchrmusum.t φ seq 1 + F T
12 dchrmusum.2 φ y 1 +∞ seq 1 + F y T C y
13 1 2 3 4 5 6 7 8 9 10 11 12 dchrisumn0 φ T 0
14 13 adantr φ x + T 0
15 ifnefalse T 0 if T = 0 log x 0 = 0
16 14 15 syl φ x + if T = 0 log x 0 = 0
17 16 oveq2d φ x + n = 1 x X L n Λ n n + if T = 0 log x 0 = n = 1 x X L n Λ n n + 0
18 fzfid φ x + 1 x Fin
19 7 ad2antrr φ x + n 1 x X D
20 elfzelz n 1 x n
21 20 adantl φ x + n 1 x n
22 4 1 5 2 19 21 dchrzrhcl φ x + n 1 x X L n
23 elfznn n 1 x n
24 23 adantl φ x + n 1 x n
25 vmacl n Λ n
26 nndivre Λ n n Λ n n
27 25 26 mpancom n Λ n n
28 24 27 syl φ x + n 1 x Λ n n
29 28 recnd φ x + n 1 x Λ n n
30 22 29 mulcld φ x + n 1 x X L n Λ n n
31 18 30 fsumcl φ x + n = 1 x X L n Λ n n
32 31 addid1d φ x + n = 1 x X L n Λ n n + 0 = n = 1 x X L n Λ n n
33 17 32 eqtrd φ x + n = 1 x X L n Λ n n + if T = 0 log x 0 = n = 1 x X L n Λ n n
34 33 mpteq2dva φ x + n = 1 x X L n Λ n n + if T = 0 log x 0 = x + n = 1 x X L n Λ n n
35 1 2 3 4 5 6 7 8 9 10 11 12 dchrvmasumif φ x + n = 1 x X L n Λ n n + if T = 0 log x 0 𝑂1
36 34 35 eqeltrrd φ x + n = 1 x X L n Λ n n 𝑂1