The totient function , also called Euler's totient function, is defined as the number of Positive Integers which are Relatively Prime to (i.e., do not contain any factor in common with) , where 1 is counted as being Relatively Prime to all numbers. Since a number less than or equal to and Relatively Prime to a given number is called a Totative, the totient function can be simply defined as the number of Totatives of . For example, there are eight Totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so .

By convention, . The first few values of for , 2, ... are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ... (Sloane's A000010). is plotted above for small .

For a Prime ,

(1) |

(2) |

(3) |

(4) |

(5) |

By induction, the general case is then

(6) |

(7) |

(8) |

The Divisor Function satisfies the Congruence

(9) |

(10) |

Walfisz (1963), building on the work of others, showed that

(11) |

(12) |

(13) | |||

(14) |

is the Möbius Function, is the Riemann Zeta Function, and is the Euler-Mascheroni Constant (Dickson). can also be written

(15) |

If the Goldbach Conjecture is true, then for every number , there are Primes and such that

(16) |

Curious equalities of consecutive values include

(17) |

(18) |

(19) |

The Summatory totient function, plotted above, is defined by

(20) |

(21) | |||

(22) |

where is the Riemann Zeta Function (Perrot 1881). The first values of are 1, 2, 4, 6, 10, 12, 18, 22, 28, ... (Sloane's A002088).

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``The Euler Totient Function.'' §24.3.2 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 826, 1972.

Beiler, A. H. Ch. 12 in *Recreations in the Theory of Numbers: The Queen of Mathematics Entertains.* New York:
Dover, 1966.

Conway, J. H. and Guy, R. K. ``Euler's Totient Numbers.'' *The Book of Numbers.* New York:
Springer-Verlag, pp. 154-156, 1996.

Courant, R. and Robbins, H. ``Euler's Function. Fermat's Theorem Again.'' §2.4.3 in Supplement to Ch. 1 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 48-49, 1996.

DeKoninck, J.-M. and Ivic, A.
*Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields.*
Amsterdam, Netherlands: North-Holland, 1980.

Dickson, L. E. *History of the Theory of Numbers, Vol. 1: Divisibility and Primality.* New York: Chelsea, pp. 113-158, 1952.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/totient/totient.html

Guy, R. K. ``Euler's Totient Function,'' ``Does Properly Divide ,'' ``Solutions of
,''
``Carmichael's Conjecture,'' ``Gaps Between Totatives,'' ``Iterations of and ,''
``Behavior of
and
.'' §B36-B42 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 90-99, 1994.

Halberstam, H. and Richert, H.-E. *Sieve Methods.* New York: Academic Press, 1974.

Honsberger, R. *Mathematical Gems II.* Washington, DC: Math. Assoc. Amer., p. 35, 1976.

Perrot, J. 1811. Quoted in Dickson, L. E. *History of the Theory of Numbers, Vol. 1: Divisibility and Primality.*
New York: Chelsea, p. 126, 1952.

Shanks, D. ``Euler's Function.'' §2.27 in *Solved and Unsolved Problems in Number Theory, 4th ed.*
New York: Chelsea, pp. 68-71, 1993.

Sloane, N. J. A. Sequences
A000010/M0299
and A002088/M1008
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Subbarao, M. V. ``On Two Congruences for Primality.'' *Pacific J. Math.* **52**, 261-268, 1974.

© 1996-9

1999-05-26