# Primes and the Lambert W function

## Abstract

**:**

## 1. Introduction

## 2. The Prime Counting Function $\mathit{\pi}\left(\mathit{x}\right)$

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**3.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Corollary**

**6.**

## 3. The $\mathit{n}$’th Prime

**Theorem**

**4.**

**Proof.**

**Corollary**

**7.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Comment:**Note that use of the Lambert W function, simply because its asymptotic expansion contains both $ln\left(x\right)$ and $ln(ln(x\left)\right)$ terms, automatically yields the first two terms of the Cesàro–Cippola asymptotic expansion [13,14]:

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

## 4. Discussion

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Lambert W Function

## References

- Corless, R.M.; Gonnet, G.H.; Hare, D.E.G.; Jeffrey, D.J.; Knuth, D.E. On the Lambert W function. Adv. Comput. Math.
**1996**, 5, 329–359. [Google Scholar] [CrossRef] - Valluri, S.R.; Jeffrey, D.J.; Corless, R.M. Some Applications of the Lambert W Function to Physics. Can. J. Phys.
**2000**, 78, 823–831. [Google Scholar] [CrossRef] - Vial, A. Fall with linear drag and Wien’s displacement law: Approximate solution and Lambert function. Eur. J. Phys.
**2012**, 33, 751. [Google Scholar] [CrossRef] - Stewart, S.M. Wien Peaks and the Lambert W Function. Rev. Bras. Ensino Física
**2011**, 33, 3308. Available online: www.sbfisica.org.br (accessed on 5 April 2018). [CrossRef] - Stewart, S.M. Spectral Peaks and Wien’s Displacement Law. J. Therm. Heat Transfer
**2012**, 26, 689–692. [Google Scholar] [CrossRef] - Valluri, S.R.; Gil, M.; Jeffrey, D.J.; Basu, S. The Lambert W function and quantum statistics. J. Math. Phys.
**2009**, 50, 102103. [Google Scholar] [CrossRef] - Boonserm, P.; Visser, M. Bounding the greybody factors for Schwarzschild black holes. Phys. Rev. D
**2008**, arXiv:0806.220978, 101502. [Google Scholar] [CrossRef] - Barkley Rosser, J. The n’th Prime is Greater than n ln n. Proc. Lond. Math. Soc.
**1938**, 45, 21–44. [Google Scholar] - Ribenboim, P. The Little Book of Big Primes; Springe: New York, NY, USA, 1991. [Google Scholar]
- Ribenboim, P. The New Book of Prime Number Records; Springer: New York, NY, USA, 1996. [Google Scholar]
- Barkley Rosser, J. Explicit Bounds for some functions of prime numbers. Am. J. Math.
**1941**, 64, 211–232. [Google Scholar] [CrossRef] - Dusart, P. The k
^{th}prime is greater than k(ln k + ln ln k − 1) for k ≥ 2. Math. Comput.**1999**, 68, 411–415. [Google Scholar] [CrossRef] - Cesàro, E. Sur une formule empirique de M. Pervouchine. Comptes rendus
**1894**, 119, 848–849. (In French) [Google Scholar] - Cipolla, M. La determinazione assintotica dell’n
^{imo}numero primo. Mat. Nap.**1902**, 3, 132–166. (In Italian) [Google Scholar] - Barkley Rosser, J.; Schoenfeld, L. Approximate Formulas for Some Functions of Prime Numbers. Ill. J. Math.
**1962**, 6, 64–97. [Google Scholar]

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**MDPI and ACS Style**

Visser, M.
Primes and the Lambert *W* function. *Mathematics* **2018**, *6*, 56.
https://doi.org/10.3390/math6040056

**AMA Style**

Visser M.
Primes and the Lambert *W* function. *Mathematics*. 2018; 6(4):56.
https://doi.org/10.3390/math6040056

**Chicago/Turabian Style**

Visser, Matt.
2018. "Primes and the Lambert *W* function" *Mathematics* 6, no. 4: 56.
https://doi.org/10.3390/math6040056