Rational approximations to gamma

The Euler-Mascheroni constant 'gamma' is defined by:

gamma = lim_(N -> oo) [ sum_(k=1)^N 1/k - ln(N) ]

Substitute N = 2^m then:

gamma = lim_(m -> oo) [ sum_(k=1)^(2^m) 1/k - m.ln(2) ]
= lim_(m -> oo) [ sum_(k=1)^(2^(m+1)) 1/k - (m+1).ln(2) ]
= lim_(m -> oo) [ sum_(k=1)^(2^m) 1/k - m.ln(2)
+ sum_(k=2^m+1)^(2^(m+1)) 1/k - ln(2) ]

Consequently: ln(2) = lim_(m -> oo) [ sum_(k=2^m+1)^(2^(m+1)) 1/k ]

Substitute: gamma = lim_(m -> oo) [ sum_(k=1)^(2^m) 1/k
- m.sum_(k=2^m+1)^(2^(m+1)) 1/k ]

Giving the limit of a _rational_ expression for 'gamma'.

We can enclose ln(2) between an upper and a lower bound, by looking
at the integral defining ln(2) as int_1^2 dx/x :

sum_(k=2^m+1)^(2^(m+1)) 1/(2^m)/(k/(2^m)) < ln(2)
sum_(k=2^m)^(2^(m+1)-1) 1/(2^m)/(k/(2^m)) > ln(2)

Resulting in:

sum_(k=2^m+1)^(2^(m+1)) 1/k < ln(2) < sum_(k=2^m)^(2^(m+1)-1) 1/k

And with help of the article "Proof of a Theorem by R.M. Young"

http://groups.google.nl/group/sci.math/msg/54df0436354854f7

we also find rational bounds for 'gamma': L < gamma < R with

L := sum_(k=1)^(2^m) 1/k - m.sum_(k=2^m)^(2^(m+1)-1) 1/k - 1/(2(2^(m))
R := sum_(k=1)^(2^m) 1/k - m.sum_(k=2^m+1)^(2^(m+1)) 1/k - 1/(2(2^m+1))

When making calculations, the _bottleneck_ is in the harmonic series,
especially [ sum_(k=2^m+1)^(2^(m+1)) 1/k ] . Half of the work can be
saved by re-using the previous summation and summing only over the odd
values of (k). I wonder of more such SAVINGS can be made when summing
the harmonic series .. ?!

Here comes a relevant snippet of code. Any comments are quite welcome.


program logaritm;

const
gamma : double = 0.577215664901532860606512090082402431042;

function twee(n : integer) : integer;
{
2^n
}
begin
twee := 1 shl n;
end;

function half(a,b : integer) : double;
{
Half of the work: odd
part of harmonic series
}
var
k,p,q,m : integer;
s : double;
begin
half := 0;
if b-a < 1 then Exit;
s := 0;
p := (a-1) div 2;
q := b div 2;
for k := p to q-1 do
begin
m := 2*k+1;
s := s + 1/m;
end;
half := s;
end;

procedure test;
{
Just a test
}
var
M : integer;
som1,som2,som3,L,R : double;
begin
som1 := 0; som2 := 1;
for M := 0 to 29 do
begin
som1 := som1 + som2;
som2 := half(twee(M)+1,twee(M+1)) + som2/2;
som3 := som2 - 1/twee(M+1) + 1/twee(M);
Writeln(M,' ',som1 - ln(2)*M,' = ',gamma);
Writeln(som2,' < ',ln(2),' < ',som3);
L := som1 - som3*M - 1/(2*twee(M));
R := som1 - som2*M - 1/(2*(twee(M)+1));
Writeln(L,' < ',gamma,' < ',R);
end;
end;

begin
test;
end.


Snippet of output:

23 5.77215724506179E-0001 = 5.77215664901533E-0001
6.93147150757633E-0001 < 6.93147180559945E-0001 < 6.93147210362278E-0001
5.77214979447887E-0001 < 5.77215664901533E-0001 < 5.77216350354724E-0001
24 5.77215694703865E-0001 = 5.77215664901533E-0001
6.93147165658792E-0001 < 6.93147180559945E-0001 < 6.93147195461114E-0001
5.77215307273491E-0001 < 5.77215664901533E-0001 < 5.77216022529230E-0001
25 5.77215679802713E-0001 = 5.77215664901533E-0001
6.93147173109328E-0001 < 6.93147180559945E-0001 < 6.93147188010490E-0001
5.77215478637945E-0001 < 5.77215664901533E-0001 < 5.77215851166975E-0001
26 5.77215672352095E-0001 = 5.77215664901533E-0001
6.93147176834559E-0001 < 6.93147180559945E-0001 < 6.93147184285140E-0001
5.77215568046453E-0001 < 5.77215664901533E-0001 < 5.77215761761548E-0001
27 5.77215668626711E-0001 = 5.77215664901533E-0001
6.93147178697210E-0001 < 6.93147180559945E-0001 < 6.93147182422500E-0001
5.77215614612439E-0001 < 5.77215664901533E-0001 < 5.77215715195277E-0001
28 5.77215666763975E-0001 = 5.77215664901533E-0001
6.93147179628528E-0001 < 6.93147180559945E-0001 < 6.93147181491174E-0001
5.77215638826938E-0001 < 5.77215664901533E-0001 < 5.77215690981002E-0001
29 5.77215665832559E-0001 = 5.77215664901533E-0001
6.93147180094224E-0001 < 6.93147180559945E-0001 < 6.93147181025547E-0001
5.77215651398791E-0001 < 5.77215664901533E-0001 < 5.77215678407146E-0001


Han de Bruijn

.

Quantcast