(1/1

^{2 }+ 1/2^{2 }+ 1/3^{2 }+ 1/4^{2 }+ …)^{2}/(1/1^{4 }+ 1/2^{4 }+ 1/3^{4 }+ 1/4^{4 }+ …)^{ }
= (π

^{2}/6)^{2}/(π^{4}/90)^{ }= 90/36 = 5/2.
And that this can be written as a Dirichlet
L-series i.e.

1/1

^{2}+ 2/2^{2}+ 2/3^{2}+ 2/4^{2}+ 2/5^{2}+ 4/6^{2}+ 2/7^{2}+ 2/8^{2}+ 2/9^{2}+ 4/10^{2}+ … = 5/2
This in turn makes use of our
generalised Riemann zeta function formula,

1/1

^{s}+ (1 + k_{1})/2^{s}+ (1 + k_{2})/3^{s}+ (1 + k_{1})/4^{s}+ (1 + k_{3})/5^{s}+ {(1 + k_{1})(1 + k_{2})}/6^{s}+ …
=
1/{1 – (1 + k

_{1})/(2^{s}+ k_{1})} * 1/{1 – (1 + k_{2})/(3^{s}+ k_{2})} * 1/{1 – (1 + k_{3})/(5^{s}+ k_{3})} * …,
where
k

_{1},_{ }k_{2}, k_{3}, …_{ }= 1.
So ζ(2)

^{2}/ζ(4) = 1/1^{2}+ 2/2^{2}+ 2/3^{2}+ 2/4^{2}+ 2/5^{2}+ 4/6^{2}+ 2/7^{2}+ 2/8^{2}+ … = 5/2
And the result can be generalised so that when s is a
positive even integer,

(1/1

^{s }+ 1/2^{s }+ 1/3^{s }+ 1/4^{s }+ …)^{2}/(1/1^{2s }+ 1/2^{2s }+ 1/3^{2s }+ 1/4^{2s }+ …)^{ }= k (where k is a rational number).
And this can be expressed as an
L1 series (with exponent s) i.e.

1/1

^{s }+ 2/2^{s }+ 2/3^{s }+ 2/4^{s }+ 2/5^{s }+ 4/6^{s }+ 2/7^{s }+ …
Here the value of numerator is
determined by the number of distinct prime factors contained by each
corresponding number in the denominator with the numerator = 2

^{t}, where t represents the number of distinct prime factors.
So 2, 3, 4 and 5 respectively
each contain one distinct prime factor. So the numerator is 2

^{1}= 2.
However 6 then contains two
distinct prime factors so the numerator in this term = 2

^{2 }= 4.
And the corresponding general
product over primes expression for this new L-series is,

1/{1 – 2/(1 + 2

^{s})} * 1/{1 – 2/(1 + 3^{s})} * 1/{1 – 2/(1 + 5^{s})} * …
Thus
to illustrate once again when s = 4, we have

(1/1

^{4 }+ 1/2^{4 }+ 1/3^{4 }+ 1/4^{4 }+ …)^{2}/(1/1^{8 }+ 1/2^{8 }+ 1/3^{8 }+ 1/4^{8 }+ …)^{ }= (π^{4}/90)^{2}/(π^{8}/9450)
= (π

^{8}/8100)/(π^{8}/9450) = 9450/8100 = 7/6
And this is represented by the
new L-series (with exponent 4) i.e.

1/1

^{4 }+ 2/2^{4 }+ 2/3^{4 }+ 2/4^{4 }+ 2/5^{4 }+ 4/6^{4 }+ 2/7^{4 }+ …,
with a corresponding product over
primes expression,

1/{1 – 2/(1 + 2

^{4})} * 1/{1 – 2/(1 + 3^{4})} * 1/{1 – 2/(1 + 5^{4})} * …
= 1/(1 – 2/17) * (1 – 2/82) * (1 – 2/626) * …

=
17/15 * 82/80 * 626/624 * …

And
again each of the individual terms (for both the sum over the integers and
product over the primes expressions) can then be expressed by a corresponding
L2 series.

For
example the first term of the product over primes expression i.e. 17/15

= 1 +
(2/17)

^{1}+ (2/17)^{2 }+ (2/17)^{3 }+ … = 17/15
And
once again an unlimited number of possible variations can be made on this basic
result through removing certain prime numbers (in the product over primes
expressions) and then the corresponding natural numbers where these are factors
in the sum over the integers expression.

So
for example if we remove 2 (in product over primes expression) and then all
corresponding even numbers (in the sum over integers expression) we have

(1/1

^{4 }+ 1/3^{4 }+ 1/5^{4 }+ 1/7^{4 }+ …)^{2}/(1/1^{8 }+ 1/3^{8 }+ 1/5^{8 }+ 1/7^{8 }+ …)^{ }= (π^{4}/96)^{2}/(255π^{8}/9450 * 256) = 35/34
And this is represented by the
new L-series expression,

1/1

^{4 }+ 2/3^{4 }+ 2/5^{4 }+ 2/7^{4 }+ 2/9^{4 }+ 2/11^{4 }+ 2/13^{4 }+ 4/15^{4 }+ …,
with a corresponding product over
primes expression

1/{1 – 2/(1 + 3

^{4})} * 1/{1 – 2/(1 + 5^{4})} * 1/{1 – 2/(1 + 7^{4})} * …
= 1/(1 – 2/82) * (1 – 2/626) * (1 – 2/2402) * …

=
82/80 * 626/624 * 2402/2400 * … = 35/34

And
just as this new L1 series represents a quotient of the two original L1 series,

in
complementary fashion, each individual terms can be expressed as a product of
two original L2 series

So
for example 82/80 = 82/81 * 81/80

= [1
+ {1/1 + 3

^{4}}^{1}+ {1/1 + 3^{4}}^{2 }+ {1/1 + 3^{4}}^{3 }+ …] * [1 + (1/3^{4}}^{1}+ (1/ 3^{4})^{2 }+ (1/3^{4})^{3 }+ …].
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