| 释义 |
Totient FunctionThe totient function  , also called Euler's totient function, is defined as the number of PositiveIntegers  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 andRelatively Prime to a given number is called a Totative, the totient function can be simply definedas 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) |
 is a Power of a Prime, thenthe numbers which have a common factor with  are the multiples of : ,  , ...,  . There are of these multiples, so the number of factors Relatively Prime to  is
 | (2) |
 be the number of Positive Integers  not Divisible by . As before, , , ...,  have common factors, so
 | (3) |
 be some other Prime dividing . The Integers divisible by are ,  ,...,  . But these duplicate  ,  , ...,  . So the number of terms which must be subtractedfrom  to obtain  is
 | (4) |
By induction, the general case is then
 | (6) |
 to ,
 | (7) |
 of to via
 | (8) |
The Divisor Function satisfies the Congruence
 | (9) |
 is theDivisor Function. No Composite solution is currently known to
 | (10) |
Walfisz (1963), building on the work of others, showed that
 | (11) |
 | (12) |
 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) |
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).See also Dedekind Function, Euler's Totient Rule, Fermat's Little Theorem, Lehmer's Problem,Leudesdorf Theorem, Noncototient, Nontotient, Silverman Constant, Totative,Totient Valence Function
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. SequencesA000010/M0299and A002088/M1008in ``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.
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