Theorem of El'Melik
Throughout this page
z(x) is equivalent to
Zeta(x); a , t any real number. 
,
| x | = Absolute value of (x)
, ln(x) = natural logarithm of (x). Following
sets of identities for the z(x)
values have been developed
recently ; In here "c" is "finite",
but it may also approach "infinity", in that case the leftmost
terms will be; |z(a
+ i.t)|2 , while rightmost terms in eqn1 and eqn2
will be; z(2.a)
.
Please note that Eqn3 is obtained by the summation of eqn1 and eqn2.
eqn1 may be controlled by a "mathematica 2.1" program
eqn2 may be controlled by a "mathematica 2.1" program
eqn3 may be controlled by a "mathematica 2.1" program
Zeta related functions
If we put; t=0 in eqn3, then we'll have the famous Euler
sums, "c"
is "finite"
Euler_sum may be controlled by a "mathematica 2.1" program
If we put; t=0, and "a" negative in eqn3, then we'll have the famous Gaussian sums, "c" is "finite"
Gaussian_sum may be controlled by a "mathematica 2.1" program Please note that the power "a" need not to be an
"integer". "a" may be any "real" number like "p" or "e".
That
means, we can compute 
OR 
as well as integer powers.
If we put; t=0, and a = (-2.m) and c = infinity in eqn3, then we'll have the famous condition z(-2.m) = 0 (m=1,2,3,4,......).
Please note that the above condition is not so easy to visualize, but it is a reality. However, If "c" is finite it can never be realised.
Relation of (p4 /120) with double summation series ;
this relation may be controlled by a mathematica 2.1 program
Relation of (p2 /6) with double summation series ;
this relation may be controlled by a mathematica 2.1 program
Relation of z(2.a) with double summation series ;
this relation may be controlled by a mathematica 2.1 program
Relation of z(a) with double summation series ;
this relation may be controlled by a mathematica 2.1 program
Relation of (z(a))2 - z(2.a) with double summation series ;
If (a = 0.6268176...) then, left side of eqn. becomes zero, hence the right hand function will also be zero.
this relation may be controlled by a mathematica 2.1 program
Some significant limiting forms of the double summation series ;
