:: Sequences of Prime Reciprocals -- Preliminaries
:: by Adam Grabowski
::
:: Received March 27, 2018
:: Copyright (c) 2018-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies ORDINAL2, FINSEQ_1, ARYTM_3, ARYTM_1, RELAT_1, FUNCT_1, INT_1,
SQUARE_1, SEQ_1, MEASURE5, FACIRC_1, FUNCT_4, ABIAN, VALUED_1, REALSET1,
SERIES_1, POWER, FUNCT_7, FINSET_1, CARD_1, MCART_1, MOEBIUS1, RCOMP_1,
SIN_COS, TAYLOR_2, INTEGRA5, PYTHTRIP, BASEL_1, LIMFUNC1, PRE_POLY,
XXREAL_2, PARTFUN1, NEWTON, XXREAL_0, ORDINAL4, XBOOLE_0, REAL_1, INT_2,
TARSKI, PARTFUN3, NAT_1, TAYLOR_1, SEQ_4, XXREAL_1, FDIFF_1, PBOOLE,
NAT_3, CARD_3, VALUED_0, UPROOTS, INT_7, SERIES_3, XCMPLX_0, SUBSET_1,
ZFMISC_1, NUMBERS, MOEBIUS2, MOEBIUS3, INTEGRA1;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, CARD_1, XTUPLE_0,
XXREAL_0, XCMPLX_0, VALUED_0, ZFMISC_1, FINSET_1, SQUARE_1, ABIAN, INT_1,
INT_2, NAT_1, XREAL_0, RELAT_1, FUNCT_1, FUNCT_2, VALUED_1, FUNCT_4,
FINSEQ_1, PARTFUN3, PRE_POLY, RCOMP_1, FCONT_1, MEASURE5, RFUNCT_1,
SEQ_4, PBOOLE, NAT_D, SEQ_1, SEQ_2, RVSUM_1, SERIES_1, PYTHTRIP, UPROOTS,
BINOP_1, NEWTON, POWER, TAYLOR_1, INT_7, SERIES_3, NAT_3, PEPIN,
MOEBIUS1, MOEBIUS2, PARTFUN1, SIN_COS, RELSET_1, PARTFUN2, FDIFF_1,
LIMFUNC1, INTEGRA5, TAYLOR_2, BASEL_1;
constructors SIN_COS, UPROOTS, INT_7, SERIES_3, COMSEQ_2, INTEGRA1, FCONT_1,
TAYLOR_2, ABIAN, MOEBIUS1, FINSOP_1, RVSUM_1, BAGORDER, SERIES_1, REAL_1,
PARTFUN3, PEPIN, NAT_3, RELSET_1, RECDEF_1, MOEBIUS2, TAYLOR_1, PYTHTRIP,
INTEGRA5, SEQ_4, PARTFUN2, BASEL_1, FDIFF_1, LIMFUNC1;
registrations RELSET_1, INT_1, XREAL_0, NAT_1, MEMBERED, FINSEQ_1, FINSET_1,
XXREAL_0, FUNCT_1, NEWTON, XCMPLX_0, NUMBERS, ORDINAL1, PRE_POLY, NAT_3,
MOEBIUS2, RCOMP_1, COMSEQ_3, INT_7, SEQ_4, XBOOLE_0, VALUED_0, XXREAL_2,
FUNCT_2, MEASURE5, ABIAN, INTEGRA1, RELAT_1, NEWTON03, XTUPLE_0,
SUBSET_1, VALUED_1, FCONT_3, POWER, NEWTON04;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
begin :: Preliminaries
reserve n,i,k,m for Nat;
reserve p for Prime;
registration
cluster non zero square non trivial for Nat;
end;
registration let Z be Subset of REAL;
let f be PartFunc of Z, REAL;
let A be Subset of REAL;
cluster f | A -> A-defined for PartFunc of Z, REAL;
end;
theorem :: MOEBIUS3:1
for Z being Subset of REAL st 0 in Z holds
(id Z)"{0} = {0};
theorem :: MOEBIUS3:2
for Z being Subset of REAL st not 0 in Z holds
(id Z)"{0} = {};
theorem :: MOEBIUS3:3
for Z being open Subset of REAL,
A being non empty closed_interval Subset of REAL st
not 0 in Z & A c= Z holds
((id Z)^) | A is continuous;
theorem :: MOEBIUS3:4
for Z being open Subset of REAL,
A being non empty closed_interval Subset of REAL
st Z = right_open_halfline 0 & A = [.1,n+1.] holds
integral ((id Z)^,A) = ln.(n + 1);
theorem :: MOEBIUS3:5
for Z being open Subset of REAL,
A being non empty closed_interval Subset of REAL
st Z = right_open_halfline 0 & 0 < n & A = [.n, n+1.] holds
integral ((id Z)^,A) = ln.((n + 1) / n);
theorem :: MOEBIUS3:6
for x,r being Real st x > 0 & r > 0 holds
Maclaurin(exp_R,].-r,r.[,x) is positive-yielding;
theorem :: MOEBIUS3:7
for f be summable Real_Sequence, n be Nat st
f is positive-yielding holds
Sum (f ^\ (n+1)) > 0;
begin :: Harmonic Numbers
:: n-th harmonic number
definition let n be Nat;
func Harmonic n -> Real equals
:: MOEBIUS3:def 1
(Partial_Sums invNAT).n;
end;
theorem :: MOEBIUS3:8
Harmonic 0 = 0;
theorem :: MOEBIUS3:9
Harmonic (n + 1) = Harmonic n + 1 / (n + 1);
theorem :: MOEBIUS3:10
Harmonic 1 = 1;
theorem :: MOEBIUS3:11
Harmonic 2 = 3 / 2;
begin :: On Exponents and Logarithms
theorem :: MOEBIUS3:12
ln.1 = 0;
theorem :: MOEBIUS3:13
for x being Real st x > 0 holds
exp_R.x > x + 1;
theorem :: MOEBIUS3:14
for x being Real st x > 0 holds
ln.(x + 1) < x;
theorem :: MOEBIUS3:15
for n being Nat st n > 0 holds
ln.((n + 1) / n) < 1 / n;
theorem :: MOEBIUS3:16
for x being Real holds
ln.(exp_R.x) = x;
theorem :: MOEBIUS3:17
for x,y being Real st 0 < x < y holds
ln.x < ln.y;
theorem :: MOEBIUS3:18
for n being non zero Nat holds
ln.(n+1) > 0;
theorem :: MOEBIUS3:19
for x,y being Real st 0 < x & 0 < y holds
ln.(x * y) = ln.x + ln.y;
theorem :: MOEBIUS3:20
for x being Real holds
ex y being non zero Nat st x < ln.(ln.(y + 1));
theorem :: MOEBIUS3:21
for A being non empty closed_interval Subset of REAL,
Z being open Subset of REAL,
n being non zero Nat st
Z = right_open_halfline 0 & A = [.n, n+1.] holds
integral ((id Z)^,A) < 1 / n;
theorem :: MOEBIUS3:22
for n being non zero Nat holds
ln.(n + 1) < Harmonic n;
theorem :: MOEBIUS3:23
for n1, n2 being Nat st n1 ^2 = n2 ^2 holds
n1 = n2;
registration let n be non trivial Nat;
cluster n ^2 -> non trivial;
end;
::$N Telescoping series
theorem :: MOEBIUS3:24
for a,b,s being Real_Sequence st
(for n being Nat holds s.n = a.n + b.n) &
(for k being Nat holds b.k = -(a.(k+1))) holds
for n being Nat holds (Partial_Sums s).n = (a.0) + (b.n);
theorem :: MOEBIUS3:25
for f1, f2 being Real_Sequence,
n being non trivial Nat st
(for k being non trivial Nat st k <= n holds f1.k < f2.k) holds
Sum (f1, n, 1) < Sum (f2, n, 1);
begin :: Some Special Sequences
definition
func Reci-seq1 -> Real_Sequence means
:: MOEBIUS3:def 2
for n being Nat holds
it.n = 1 / (n ^2 - 1 / 4);
end;
theorem :: MOEBIUS3:26
for n being Nat holds
Reci-seq1.n = 1 / (n - 1 / 2) - 1 / (n + 1 / 2);
theorem :: MOEBIUS3:27
Reci-seq1 = (rseq (0,1,1, -1/2)) + (- rseq (0,1,1, 1 / 2));
theorem :: MOEBIUS3:28
for n being Nat holds
(Partial_Sums Reci-seq1).n < -2;
theorem :: MOEBIUS3:29
for n being Nat holds
Sum (Reci-seq1, n, 1) < 2 / 3;
registration
cluster Basel-seq -> summable;
end;
theorem :: MOEBIUS3:30
for n being Nat holds
(Partial_Sums Reci-seq1).n = -2 + - 1 / (n + 1 / 2);
theorem :: MOEBIUS3:31
for n being non trivial Nat holds
Sum (Basel-seq, n, 1) < Sum (Reci-seq1, n, 1);
theorem :: MOEBIUS3:32
for n being non trivial Nat holds
Sum (Basel-seq, n) < 5 / 3;
theorem :: MOEBIUS3:33 ::: Similar bound in BASEL series, counted independently
(Partial_Sums Basel-seq).n < 5 / 3;
definition
func Reci-seq2 -> Real_Sequence means
:: MOEBIUS3:def 3
for n being Nat holds
it.n = 1 + 1 / primenumber n;
end; :: similar functor is in MOEBIUS2
theorem :: MOEBIUS3:34
Sum Sgm {1} = 1;
definition let n be Nat;
func SetPrimes n -> Subset of NAT equals
:: MOEBIUS3:def 4
SetPrimes /\ Seg n;
end;
registration let n be Nat;
cluster SetPrimes n -> finite;
end;
theorem :: MOEBIUS3:35
for m, n being Nat st m <= n holds
SetPrimes m c= SetPrimes n;
theorem :: MOEBIUS3:36
n+1 is not Prime implies SetPrimes (n+1) = SetPrimes n;
theorem :: MOEBIUS3:37
SetPrimes 0 = {} & SetPrimes 1 = {};
theorem :: MOEBIUS3:38
n+1 is Prime implies SetPrimes (n+1) = SetPrimes n \/ {n+1};
theorem :: MOEBIUS3:39
for p being Prime st p > 2 holds
p+1 is not Prime;
theorem :: MOEBIUS3:40
SetPrimes 2 = {2};
theorem :: MOEBIUS3:41
not n+1 in SetPrimes n;
definition let n be Nat;
:: just to get an index for appropriate sequence of primes
func indexp n -> Nat equals
:: MOEBIUS3:def 5
card SetPrimes n;
end;
theorem :: MOEBIUS3:42
for n being Nat holds
indexp n <= n;
begin :: Squarefree and Square-containing Parts of a Natural Number
theorem :: MOEBIUS3:43
for n being non zero Nat holds
n = (TSqF n) * (n div TSqF n);
theorem :: MOEBIUS3:44 ::: MOEBIUS2:45 strenghtened
for n being non zero Nat holds (SqF n) |^ 2 divides n;
theorem :: MOEBIUS3:45
for m being finite-support natural-valued ManySortedSet of SetPrimes,
p being Prime st support m = {p} holds
Product m = m.p;
theorem :: MOEBIUS3:46
for n being non zero Nat holds (SqF n) |^ 2 = TSqF n;
registration
let n be non zero Nat;
cluster n div ((SqF n) |^ 2) -> square-free for Nat;
end;
::: TSqF n should be revised to allow zero for an argument
definition let n be non zero Nat;
func SquarefreePart n -> non zero Nat equals
:: MOEBIUS3:def 6
n div TSqF n;
end;
registration let n be non zero Nat;
cluster SquarefreePart n -> square-free;
end;
theorem :: MOEBIUS3:47 ::: squarefree-decompose
for n being non zero Nat holds
n = (SquarefreePart n) * (SqF n) ^2;
theorem :: MOEBIUS3:48
for n being non zero Nat holds
support pfexp n c= Seg n;
theorem :: MOEBIUS3:49 ::: a kind of generalized MOEBIUS1:14
for n being non zero Nat holds
support ppf n c= Seg n;
theorem :: MOEBIUS3:50
for n being non zero Nat holds
Seg SquarefreePart n c= Seg n;
theorem :: MOEBIUS3:51
for k,n being non zero Nat holds
k^2 divides SquarefreePart n iff k = 1;
theorem :: MOEBIUS3:52
for m,n being non zero Nat st
SquarefreePart n = SquarefreePart m &
TSqF m = TSqF n holds
m = n;
begin :: Generating Bags from Subsets of Prime Numbers
definition let A be finite Subset of SetPrimes;
func A-bag -> bag of SetPrimes equals
:: MOEBIUS3:def 7
(EmptyBag SetPrimes) +* (id A);
end;
theorem :: MOEBIUS3:53
for A being finite Subset of SetPrimes holds
support (A-bag) = A;
theorem :: MOEBIUS3:54
for A being finite Subset of SetPrimes st A = {} holds
A-bag = EmptyBag SetPrimes;
theorem :: MOEBIUS3:55
for A being finite Subset of SetPrimes
for i being object st i in support (A-bag) holds
(A-bag).i = i;
theorem :: MOEBIUS3:56
for A,B be finite Subset of SetPrimes st
A-bag = B-bag holds
A = B;
registration let A be finite Subset of SetPrimes;
cluster A-bag -> prime-factorization-like;
end;
registration let A be finite Subset of SetPrimes;
cluster Product (A-bag) -> square-free for Nat;
end;
theorem :: MOEBIUS3:57
for n being non zero Nat
for x being object st x in bool SetPrimes n holds
x is finite Subset of SetPrimes;
theorem :: MOEBIUS3:58
for n being non zero Nat
for x being object st x in [:bool SetPrimes n, Seg n:] holds
x`1 is finite Subset of SetPrimes;
:: Later we should show the connection between sum and products of
:: sequences with exponential function, respectively.
theorem :: MOEBIUS3:59
rseq(0,1,1,0) = invNAT;
theorem :: MOEBIUS3:60
indexp 0 = 0;
theorem :: MOEBIUS3:61
for n being Nat holds
(Partial_Product Reci-seq2).n > 0;
theorem :: MOEBIUS3:62
for n being Nat holds
ln.((Partial_Product Reci-seq2).n) <= (Partial_Sums ReciPrime).n;
theorem :: MOEBIUS3:63
for n being Nat holds
ln.((Partial_Product Reci-seq2).(indexp n)) <= (Partial_Sums ReciPrime).n;
definition
func Reci-Sqf-> Real_Sequence means
:: MOEBIUS3:def 8
it.0 = 0 &
for i being non zero Nat holds it.i = 1 / SquarefreePart i;
func Reci-TSq -> Real_Sequence means
:: MOEBIUS3:def 9
it.0 = 0 &
for i being non zero Nat holds it.i = 1 / TSqF i;
end;
theorem :: MOEBIUS3:64
rseq (0,1,1,0) = Reci-Sqf (#) Reci-TSq;
reserve s, s1, s2 for Real_Sequence;
theorem :: MOEBIUS3:65
for n being Nat holds Reci-Sqf.n >= 0;
theorem :: MOEBIUS3:66
for n being Nat holds Reci-TSq.n >= 0;
theorem :: MOEBIUS3:67
for n being Nat holds Basel-seq.n >= 0;
theorem :: MOEBIUS3:68 :::Seqs5:
for n being Nat holds
(Partial_Sums rseq (0,1,1,0)).n <=
(Partial_Sums Reci-Sqf).n * (Partial_Sums Reci-TSq).n;
definition let n be non zero Nat;
func Compose n -> Function of [:bool SetPrimes n, Seg n:], NAT means
:: MOEBIUS3:def 10
for x being Element of [:bool SetPrimes n, Seg n:],
A being finite Subset of SetPrimes,
k being Nat st x = [A, k] holds
it.x = (Product ((A,1)-bag)) * k ^2;
end;
theorem :: MOEBIUS3:69
(Partial_Sums Basel-seq).n >= 0;
begin :: On Reciprocals of Products of Prime Numbers
definition let n be Nat;
func ReciProducts n -> Subset of REAL equals
:: MOEBIUS3:def 11
the set of all 1 / Product Sgm X where X is Subset of SetPrimes n;
end;
registration let n be Nat;
cluster ReciProducts n -> finite;
end;
theorem :: MOEBIUS3:70
ReciProducts 0 = { 1 };
theorem :: MOEBIUS3:71
for p being Prime st p > 2 holds
ReciProducts (p+1) = ReciProducts p;
theorem :: MOEBIUS3:72
for p being Nat st p+1 is not Prime holds
ReciProducts (p+1) = ReciProducts p;
theorem :: MOEBIUS3:73
ReciProducts 1 = { 1 };
theorem :: MOEBIUS3:74
ReciProducts 2 = { 1 / 2, 1 };
theorem :: MOEBIUS3:75
for n be Nat holds
ReciProducts n c= ReciProducts (n+1);
theorem :: MOEBIUS3:76
for n be Nat st n+1 is Prime holds
ReciProducts (n+1) = ReciProducts n \/
{1 / Product Sgm X where X is Subset of SetPrimes (n+1) : n+1 in X};
theorem :: MOEBIUS3:77
for n be Nat st n+1 is Prime holds
ReciProducts (n+1) =
{1 / Product Sgm X where X is Subset of SetPrimes n :
not n+1 in X}
\/ {1 / Product Sgm X where X is Subset of SetPrimes (n+1) :
n+1 in X};