A family of rational zeta series

Eric Dolores-Cuenca
Finance Fishery Manufacture Industrial
 Mathematics Center on Big Data 
 Pusan National University

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Table of contents

The identity
How we found it?
References

The identity

Rational zeta series

Let   denote the Riemann zeta function. 
 We know  is irrational, we know that at least one out of   is irrational, and at least two out of   are irrational.

Series

Christian Goldbach (1690-1764) discovered that
 . 
 Series in which the n term is a rational multiplied by  are called rational zeta series by [1].

The main identity

In [1] the following identity was discovered:

Define the Hurwitz zeta function by 

 we computed the following formula, for :

Example

When a=2 we obtain:
 

Because of the infinite number of terms of the form  , the left hand side of this identity cannot be computed as of January 2013.

How we found it?

Operad theory

We study associative operations. 
The sum of two numbers is represented by a tree:

Associativity

Associativity of the sum implies that the following two trees return the same value:

Composition

By using the grafting of trees we can encode composition:

Operad theory

A (non symmetrical) operad of sets is a family  together with associative composition rules:
 
 In the example above, the operad of trees has as n-ary operations trees with n leaves, and composition is grafting of trees.

The operad of posets

A poset is a set with a partial order. For example: consider a list of programs, we say  if the output of  is needed to run . Then a poset help us understand the order to run the programs.

The operad of posets

Examples of posets:

 is the poset union of  and  and no new order. 
Two programs can be run in parallel if they don’t require the output of the other program.
 is the poset where the maximum elements of  are forced to be smaller than the minimum elements of . 

Then  stands for three programs, in which the output of  and  is needed to run .

The operad of posets

We work with the following operad:
For every ,  contains trees with  leaves, and every vertex with   children is labeled with a partially ordered set with  points. 

Example of operation in :

The operad of posets

For every set  we have  the operad with .

An algebra  over an operad  is a map 

the abstract operations of  become functions on .

How does an algebra over the operad of poset looks like?

Given a poset  with  points, the order polytope  is a subset of  where the points satisfy the inequalities of the poset .
 (a triangle).

 (a square).
 is the image below:

How does an algebra over the operad of poset looks like?

Given posets  , define the action of  on polytopes to be the Minkovski sum of the polytopes:

For example :
 is the figure below:

How does an algebra over the operad of poset looks like?

Given posets _1,Q_2 , 


because posets are an algebra over the operad , 
and polytopes are an algebra over the operad ,
 and there is an operadic homomorphism 
,
We claim that any poset with  points, defines an -ary operation, and

How did we find the identity?

In [2] we constructed several algebras over the operad of posets, and constructed homomorphism among them:

The vertical arrow from polynomials to zeta values is equivalent to:

References

[1] J. M. Borwein, D. M. Bradley, and R. E. Crandall. Computational strategies for the Riemann zeta function. Journal of Computational and Applied Mathematics, 121(1):247–296, 2000.

[2] E. Dolores - Cuenca, J. L . Mendoza - Cortes. A poset version of Ramanujan results on Eulerian numbers and zeta values. Preprint, 2023 arxiv=2205.05208.