# Combinatorial Models of the Distribution of Prime Numbers

## Abstract

**:**

## 1. Introduction: The Bingo Bag of Primes

## 2. First Occurrence Sequences: A New Class of Combinatorial Objects

#### 2.1. Definitions and Probability Spaces

**Remark**

**1.**

**Remark**

**2.**

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Definition**

**4.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Remark**

**6.**

**Remark**

**7.**

#### 2.2. Probability Relations

**Notation.**

**Assumption**

**1.**

**Lemma**

**1.**

**Corollary**

**2.**

**Proof.**

## 3. Combinatorics of FOS

#### 3.1. Ordered FOS

**Definition**

**5.**

**Example**

**1.**

**Remark**

**8.**

**Theorem**

**2.**

**Proof.**

#### 3.2. Counting Ordered FOS

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.3. Ordered FOS and Stirling Numbers of the Second Kind

**Theorem**

**5.**

**Remark**

**9.**

## 4. First Occurrence Sequences, Set Partitions, and the Sequence of Primes

#### 4.1. Ordered FOS and the Prime Number Theorem

#### 4.2. FOS Probabilities and Some Combinatorial Identities Involving Stirling Numbers

**Corollary**

**5.**

**Corollary**

**6.**

**Proof**

**of**

**Equation**

**(73).**

**Remark**

**10.**

**Remark**

**11.**

#### 4.3. Bounds of $g(k,i)$ Numbers and Stirling Numbers of the Second Kind

**Definition**

**6.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Remark**

**12.**

**Remark**

**13.**

#### 4.4. FOS Average Length and Harmonic Numbers

**Lemma**

**6.**

**Lemma**

**7.**

**Corollary**

**7.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**8.**

#### 4.5. FOS as a Model of the Distribution of Primes

**Remark**

**14.**

**Definition**

**7.**

**Definition**

**8.**

**Theorem**

**8.**

**Proof.**

**Remark**

**15.**

**Remark**

**16.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 5. Model Generalization and the Connection with the Riemann Hypothesis

#### 5.1. Model Generalization

**Notation.**

**Remark**

**17.**

**Remark**

**18.**

**Remark**

**19.**

**Theorem**

**11.**

**Remark**

**20.**

**Lemma**

**8.**

**Proof.**

**Theorem**

**12.**

**Proof.**

#### 5.2. Counting Successive Prime Pairs through the log(n)-Model

**Remark**

**21.**

**Definition**

**9.**

**Assumption**

**2.**

#### 5.3. A Heuristic Model of the Distribution of Prime Numbers

**Remark**

**22.**

**Definition**

**10.**

**Theorem**

**13.**

**Proof.**

**Remark**

**23.**

## 6. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numbers of successive prime pairs as function of the gap $d={p}_{n+1}-{p}_{n}$ in 49,000,000 $\le n\le $ 50,000,000 (logarithmic scale).

**Figure 2.**Numbers of successive prime pairs as function of the gap $d={p}_{n+1}-{p}_{n}$ in 4,461,632,979,716 $\le n\le $ 8,731,188,863,469 (logarithmic scale).

n | ${\mathit{L}}_{\mathit{n}/\mathit{n}}^{\prime}$ | ${\mathit{p}}_{\mathit{n}}$ |
---|---|---|

2 | 3 | 3 |

3 | 5.5 | 5 |

4 | 8.333 | 7 |

5 | 11.416 | 11 |

6 | 14.7 | 13 |

$\mathit{j}=1$ | $\mathit{j}=2$ | $\mathit{j}=3$ | $\mathit{j}=4$ | $\mathit{j}=5$ | $\mathit{j}=6$ | |
---|---|---|---|---|---|---|

$i=3$ | 1 | |||||

$i=4$ | $-\frac{1}{2}$ | 2 | ||||

$i=5$ | $\frac{1}{6}$ | $-2$ | $\frac{9}{2}$ | |||

$i=6$ | $-\frac{1}{24}$ | $\frac{4}{3}$ | $-\frac{27}{4}$ | $\frac{32}{3}$ | ||

$i=7$ | $\frac{1}{120}$ | $-\frac{2}{3}$ | $\frac{27}{4}$ | $-\frac{64}{3}$ | $\frac{625}{24}$ | |

$i=8$ | $-\frac{1}{720}$ | $\frac{4}{15}$ | $-\frac{81}{16}$ | $\frac{256}{9}$ | $-\frac{3125}{48}$ | $\frac{324}{5}$ |

k | $\mathit{\pi}(\mathit{k})$ | $\mathit{li}(\mathit{k})$ | $\widehat{\mathit{\pi}}(\mathit{k})={\sum}_{\mathit{n}=1}^{\mathit{k}}\mathit{w}(\mathit{k},\mathit{n})$ | $\widehat{\mathit{\pi}}(\mathit{k})={\int}_{1}^{\mathit{k}}{\mathit{e}}^{-{\mathit{ane}}^{-\frac{\mathit{k}}{\mathit{n}}}}\mathit{dn}$ |
---|---|---|---|---|

100 | 25 | $30.1261$ | $26.9462$ | $23.1623$ |

200 | 46 | $50.1921$ | $47.0309$ | $41.3716$ |

300 | 62 | $68.3336$ | $65.50659$ | $58.2265$ |

400 | 78 | $85.4178$ | $83.06021$ | $74.2987$ |

500 | 95 | $101.7939$ | $99.98131$ | $89.8312$ |

600 | 109 | $117.6465$ | $116.4275$ | $104.9572$ |

700 | 125 | $133.0889$ | $132.4971$ | $119.7596$ |

800 | 139 | $148.1967$ | $148.2565$ | $134.2952$ |

900 | 154 | $163.0236$ | $163.7537$ | $148.6046$ |

1000 | 168 | $177.6097$ | $179.0246$ | $162.7185$ |

2000 | 303 | $314.8092$ | $323.4725$ | $296.6907$ |

3000 | 430 | $442.7592$ | $458.9438$ | $422.8337$ |

4000 | 550 | $565.3645$ | $589.1406$ | $544.3509$ |

5000 | 669 | $684.2808$ | $715.6488$ | $662.6328$ |

6000 | 783 | $800.4141$ | $778.4514$ | |

7000 | 900 | $914.3308$ | $892.2944$ | |

8000 | 1007 | $1026.416$ | $1004.4960$ | |

9000 | 1117 | $1136.949$ | $1115.2989$ | |

10,000 | 1229 | $1246.137$ | $1224.8866$ |

k | $\mathit{\pi}(\mathit{k})$ | $\widehat{\mathit{\pi}}(\mathit{k})={\int}_{1}^{\mathit{k}}{\mathit{e}}^{-{\mathit{a}\mathit{n}\mathit{e}}^{-\frac{\mathit{k}}{\mathit{n}}}}\mathit{d}\mathit{n}$ | $\widehat{\mathit{\pi}}(\mathit{k})/\mathit{\pi}(\mathit{k})$ |
---|---|---|---|

${10}^{5}$ | 9592 | $9428.02$ | $0.98291$ |

${10}^{6}$ | 78,498 | 78,480.93 | $0.99978$ |

${10}^{7}$ | 664,579 | 671,099.45 | $1.00981$ |

${10}^{8}$ | 5,761,455 | 5,855,689.76 | $1.01636$ |

${10}^{9}$ | 50,847,534 | 51,900,660.41 | $1.02071$ |

${10}^{10}$ | 455,052,511 | 465,792,892.49 | $1.02360$ |

${10}^{11}$ | 4,118,054,813 | 4,223,145,802.17 | $1.02552$ |

${10}^{12}$ | 37,607,912,018 | 38,614,679,105.06 | $1.02677$ |

${10}^{13}$ | 346,065,536,839 | 355,603,668,431.86 | $1.02756$ |

${10}^{14}$ | 3,204,941,750,802 | 3,294,779,143,238.6 | $1.02803$ |

${10}^{15}$ | 29,844,570,422,669 | 30,688,289,307,555 | $1.02827$ |

${10}^{16}$ | 279,238,341,033,925 | 287,153,196,808,146 | $1.02834$ |

${10}^{17}$ | 2,623,557,157,654,233 | $2.69779945552531\times {10}^{15}$ | $1.02830$ |

${10}^{18}$ | 24,739,954,287,740,860 | $2.54367476772712\times {10}^{16}$ | $1.02816$ |

${10}^{19}$ | 234,057,667,276,344,607 | $2.40603623244741\times {10}^{17}$ | $1.02797$ |

${10}^{20}$ | 2,220,819,602,560,918,840 | $2.28238863108907\times {10}^{18}$ | $1.02772$ |

${10}^{21}$ | 21,127,269,486,018,731,928 | $2.17071405049150\times {10}^{19}$ | $1.02745$ |

${10}^{22}$ | 201,467,286,689,315,906,290 | $2.06936330707283\times {10}^{20}$ | $1.02715$ |

${10}^{23}$ | 1,925,320,391,606,803,968,923 | $1.97697565027884\times {10}^{21}$ | $1.02683$ |

${10}^{24}$ | 18,435,599,767,349,200,867,866 | $1.89241852576407\times {10}^{22}$ | $1.02650$ |

${10}^{25}$ | 176,846,309,399,143,769,411,680 | $1.81474179606716\times {10}^{23}$ | $1.02617$ |

${10}^{26}$ | 1,699,246,750,872,437,141,327,603 | $1.74314253336239\times {10}^{24}$ | $1.02583$ |

${10}^{27}$ | 16,352,460,426,841,680,446,427,399 | $1.676937646378480\times {10}^{25}$ | $1.02550$ |

k | $\mathit{\pi}(\mathit{k})$ | $\widehat{\mathit{\pi}}(\mathit{k})$ |
---|---|---|

10 | 4 | $4.33$ |

100 | 25 | $25.50$ |

1000 | 168 | $167.65$ |

10,000 | 1229 | $1229.34$ |

100,000 | 9592 | 9592.09 |

1,000,000 | 78,498 | 78,498.23 |

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Barbarani, V. Combinatorial Models of the Distribution of Prime Numbers. *Mathematics* **2021**, *9*, 1224.
https://doi.org/10.3390/math9111224

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Barbarani V. Combinatorial Models of the Distribution of Prime Numbers. *Mathematics*. 2021; 9(11):1224.
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**Chicago/Turabian Style**

Barbarani, Vito. 2021. "Combinatorial Models of the Distribution of Prime Numbers" *Mathematics* 9, no. 11: 1224.
https://doi.org/10.3390/math9111224