More Connections

It struck me clearly the other day that each L2 function can be expressed as the quotient of two L1 functions.

For example in the Riemann zeta function (where s = 2) the first term in the product over primes expression is 4/3, which is related to the corresponding first prime 2, through the relationship 
1 – 1/2².

Now this can be expressed as the quotient of the functions,

1/1² + 1/2² + 1/3²  + 1/5² + … and 1/1²  + 1/3² + 1/5²  + 1/7² + … (where in the latter case all numbers which contain 2 as a shared factor are omitted).

So (1/1² + 1/2² + 1/3²  + 1/5² + …)/(1/1²  + 1/3² + 1/5²  + 1/7² + …)

= 1 + (1/2²)¹ + (1/2²)² + (1/2²)³ + …    = 4/3


And notice the complementarity here! Whereas the natural numbers represent the base aspect in the L1, they represent the dimensional aspect of number in the L2; and whereas 2 is constant as a dimensional number in the L1 it is constant (as reciprocal) in L2. So this properly signifies the dynamic interactive relationship connecting both functions.

By definition, any of the individual terms (in the product over primes expressions) or indeed combination of terms - which can be expressed as an L2 function or product of L2 functions - can be expressed as the quotient of two L1 functions.

For example the second term in the Riemann zeta function (where s = 3) is 27/26 relating to 1 – 1/3³.

This can be expressed as,

(1/1³ + 1/2³ + 1/3³ + 1/4³ + …)/(1/1³  + 1/2³ + 1/4³ + 1/5²  + …)

= 1 + (1/3³)¹ + (1/3³)² + (1/3³)³ + …    = 27/26

And if for example the first two terms 8/7 and 27/26 relating to the primes 2 and 3 respectively, were to be omitted, we would have,

(1/1³ + 1/2³ + 1/3³ + 1/5³ + …)/(1/1³  + 1/5³ + 1/7³ + 1/11³  + …)

= {1 + (1/2³)¹ + (1/2³)² + (1/2³)³ + …} * {1 + (1/3³)¹ + (1/3³)² + (1/3³)³ + …}

I mentioned in a previous blog entry that the generalised expression for the Riemann zeta function linking sum over the integers and product over the primes expressions is

1/1^s + (1 + k₁)/2^s + (1 + k₂)/3^s + (1 + k₁)/4^s + (1 + k₃)/5^s + {(1 + k₁)(1 + k₂)}/6^s + …
= 1/{1 – (1 + k₁)/(2^s + k₁)} * 1/{1 – (1 + k₂)/(3^s + k₂)} * 1/{1 – (1 + k₃)/(5^s + k₃)} * …,
 

Then when k₁ = k₂ = k₃,… = 1 that when s is an even number that a rational value results for the corresponding expressions.

So again when s = 2 and k₁ = k₂ = k₃,… = 1, then

1/1² + 2/2² + 2/3² + 2/4² + 2/5² + 4/6² + 2/7² + 2/8² + 2/9² + 4/10² + …

= 5/3 * 10/8 * 26/24 * 50/48 * …    = 5/2

Now if we eliminate any of the individual product over primes terms, then a resulting Dirichlet L-series will again result with a rational number value.

So if we eliminate the first term (in the product over primes expression) i.e. 5/3 (relating to 2) then we have,

1/1² + 2/3² + 2/5² + 2/7² + 2/9² + 2/11² + … = 10/8 * 26/24 * 50/48 * 122/120 * … = 3/2

And if alternatively we eliminate the second term (in the product over primes expression) i.e. 10/8 (relating to 3) we have

1/1² + 2/2² + 2/4² + 2/7² + 2/8² + 2/10² + … = 5/3 * 26/24 * 50/48 * 122/120 …   = 2