In[]:=

ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{100},10,"StatesGraph"]

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In[]:=

ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{1000},10,"StatesGraph",AspectRatio1/2]

Out[]=

In[]:=

TransitiveReductionGraph[%]

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In[]:=

TransitiveReductionGraph[ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{5040},10,"StatesGraph",AspectRatio1/2]]

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In[]:=

Graph3D[%]

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Only need to go 2 steps...

In[]:=

TransitiveReductionGraph[ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{5040},2,"StatesGraph",AspectRatio1/2]]

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In[]:=

PrimePi[1000]

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168

Total number of nodes in the graph is number of divisors:

In[]:=

DivisorSigma[0,1000]

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16

Total “weight” of graph is

In[]:=

DivisorSigma[1,1000]

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2340

#### Edge count

Edge count

In[]:=

Table[EdgeCount[ResourceFunction["MultiwayFunctionSystem"][nMost[Divisors[n]],{n},2,"StatesGraph"]],{n,100}]

Out[]=

{0,1,1,3,1,5,1,6,3,5,1,12,1,5,5,10,1,12,1,12,5,5,1,22,3,5,6,12,1,19,1,15,5,5,5,27,1,5,5,22,1,19,1,12,12,5,1,35,3,12,5,12,1,22,5,22,5,5,1,42,1,5,12,21,5,19,1,12,5,19,1,48,1,5,12,12,5,19,1,35,10,5,1,42,5,5,5,22,1,42,5,12,5,5,5,51,1,12,12,27}

In[]:=

Table[PrimePi[n],{n,100}]

Out[]=

{0,1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,9,9,9,9,10,10,11,11,11,11,11,11,12,12,12,12,13,13,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,18,18,18,18,18,18,19,19,19,19,20,20,21,21,21,21,21,21,22,22,22,22,23,23,23,23,23,23,24,24,24,24,24,24,24,24,25,25,25,25}

In[]:=

ListLinePlot[{%97,%98}]

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In[]:=

Table[DivisorSigma[0,n],{n,100}]

Out[]=

{1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9}

In[]:=

ListLinePlot[%]

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Related to Riemann hypothesis

https://en.wikipedia.org/wiki/Divisor_function

https://mathworld.wolfram.com/RobinsTheorem.html

https://mathworld.wolfram.com/RobinsTheorem.html

#### Relation between spectral properties of “Hamiltonian” and cycle-like structure in MW graph

Relation between spectral properties of “Hamiltonian” and cycle-like structure in MW graph

#### What is the transitive reduction doing at each node?

What is the transitive reduction doing at each node?

Dividing by the prime factors.....

Depth is determined by largest exponent in prime factorization.....

Prime factorization is trying to find the predecessors of 1...

The presence of multiplication in the number is like the concatenation of rules to the multiway system

Number of distinct prime factors gives dimension....

### Analogy with harmonic oscillator

Analogy with harmonic oscillator

### Causal edges

Causal edges

[[ In the code, a causal connection is made between shared prime factors ]]

[ Creator and destroyer events of prime factors ... ]

[ Creator and destroyer events of prime factors ... ]

## Branchial structure

Branchial structure

[[ Related to equivalence of binary quadratic forms ???? ]]

## Direct Riemann zeta

Direct Riemann zeta

## Interpreting multiway systems as having real and imaginary values

Interpreting multiway systems as having real and imaginary values

The coordinatization of branchial space is then the assignment of a phase

###

#### Path weights are determined by the multiplicities of factors, so e.g. the final primes have weights that are their exponents in the factorization

Path weights are determined by the multiplicities of factors, so e.g. the final primes have weights that are their exponents in the factorization

## Real vs complex integer functions

Real vs complex integer functions

This is a pure real function iteration:

A “complete superposition” of a particular branchlike hypersurface can be thought of as a sum of the complex numbers obtained from state weights and phases

Across Pascal’s triangle, we are assigning each number a complex phase

#### Imagine that every node has some small vector of integers [or just a single integer]

Imagine that every node has some small vector of integers [or just a single integer]

### What is the analog between choice of branch cuts and choice of foliations?

What is the analog between choice of branch cuts and choice of foliations?

Each sheet in Riemann surface is like a hypersurface

## [ Something different : ]

[ Something different : ]

#### When it branches, it’s like e.g. solving for something, or taking a square root

When it branches, it’s like e.g. solving for something, or taking a square root

(For the divisor multiway function, it’s like “solving for what divides”)

#### Claim: any multiway system graph can be generated from a complex function iteration

Claim: any multiway system graph can be generated from a complex function iteration