The theorem giving an asymptotic form for the Prime Counting Function for number of Primes less than
some Integer . Legendre (1808) suggested that, for large ,

(1) |

(2) |

(3) |

For small , it has been checked and always found that . However, Skewes proved that the first crossing of occurs before (the Skewes Number). The upper bound for the crossing has subsequently been reduced to . Littlewood (1914) proved that the Inequality reverses infinitely often for sufficiently large (Ball and Coxeter 1987). Lehman (1966) proved that at least reversals occur for numbers with 1166 or 1167 Decimal Digits.

Chebyshev (Rubinstein and Sarnak 1994) put limits on the Ratio

(4) |

(5) |

Hadamard and Vallée Poussin proved the prime number theorem by showing that the Riemann Zeta Function has
no zeros of the form (Smith 1994, p. 128). In particular, Vallée Poussin showed that

(6) |

Riemann estimated the Prime Counting Function with

(7) |

(8) |

The prime number theorem is equivalent to

(9) |

The Riemann Hypothesis is equivalent to the assertion that

(10) |

(11) | |||

(12) |

Ramanujan showed that for sufficiently large ,

(13) |

(14) |

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.*
New York: Dover, pp. 62-64, 1987.

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, 1994.

Courant, R. and Robbins, H. ``The Prime Number Theorem.'' §1.2c in Supplement to Ch. 1 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 27-30, 1996.

Davenport, H. ``Prime Number Theorem.'' Ch. 18 in *Multiplicative Number Theory, 2nd ed.*
New York: Springer-Verlag, pp. 111-114, 1980.

de la Vallée Poussin, C.-J. ``Recherches analytiques la théorie des nombres premiers.'' *Ann. Soc. scient. Bruxelles* **20**,
183-256, 1896.

Hadamard, J. ``Sur la distribution des zéros de la fonction et ses conséquences arithmétiques (').''
*Bull. Soc. math. France* **24**, 199-220, 1896.

Hardy, G. H. and Wright, E. M. ``Statement of the Prime Number Theorem.'' §1.8 in
*An Introduction to the Theory of Numbers, 5th ed.* Oxford, England: Clarendon Press, pp. 9-10, 1979.

Ingham, A. E. *The Distribution of Prime Numbers.* London: Cambridge University Press, p. 83, 1932.

Legendre, A. M. *Essai sur la Théorie des Nombres.* Paris: Duprat, 1808.

Lehman, R. S. ``On the Difference
.'' *Acta Arith.* **11**, 397-410, 1966.

Littlewood, J. E. ``Sur les distribution des nombres premiers.'' *C. R. Acad. Sci. Paris* **158**, 1869-1872, 1914.

Nagell, T. ``The Prime Number Theorem.'' Ch. 8 in *Introduction to Number Theory.* New York: Wiley, 1951.

Riemann, G. F. B. ``Über die Anzahl der Primzahlen unter einer gegebenen Grösse.''
*Monatsber. Königl. Preuss. Akad. Wiss. Berlin*, 671, 1859.

Rubinstein, M. and Sarnak, P. ``Chebyshev's Bias.'' *Experimental Math.* **3**, 173-197, 1994.

Selberg, A. and Erdös, P. ``An Elementary Proof of the Prime Number Theorem.'' *Ann. Math.* **50**, 305-313, 1949.

Shanks, D. ``The Prime Number Theorem.'' §1.6 in *Solved and Unsolved Problems in Number Theory, 4th ed.*
New York: Chelsea, pp. 15-17, 1993.

Smith, D. E. *A Source Book in Mathematics.* New York: Dover, 1994.

Wagon, S. *Mathematica in Action.* New York: W. H. Freeman, pp. 25-35, 1991.

© 1996-9

1999-05-26