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1: 27.1 Special Notation
§27.1 Special Notation
d , k , m , n

positive integers (unless otherwise indicated).

2: 27.3 Multiplicative Properties
§27.3 Multiplicative Properties
27.3.8 ϕ ( m ) ϕ ( n ) = ϕ ( m n ) ϕ ( ( m , n ) ) / ( m , n ) .
3: 27.4 Euler Products and Dirichlet Series
27.4.1 n = 1 f ( n ) = p ( 1 + r = 1 f ( p r ) ) ,
Euler products are used to find series that generate many functions of multiplicative number theory. …
27.4.4 F ( s ) = n = 1 f ( n ) n - s ,
4: 27.12 Asymptotic Formulas: Primes
5: 27.2 Functions
§27.2(i) Definitions
27.2.8 a ϕ ( n ) 1 ( mod n ) ,
6: 27.13 Functions
Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …
7: Bibliography D
  • H. Davenport (2000) Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.
  • 8: 27.8 Dirichlet Characters
    §27.8 Dirichlet Characters
    An example is the principal character (mod k ): … If ( n , k ) = 1 , then the characters satisfy the orthogonality relationA divisor d of k is called an induced modulus for χ if … If k is odd, then the real characters (mod k ) are the principal character and the quadratic characters described in the next section.
    9: 27.22 Software
  • Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below 10 16 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard p - 1 , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

    For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

  • Number Theory Web. References and links to software for factorization and primality testing.

  • 10: Bibliography
  • A. Adelberg (1996) Congruences of p -adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.
  • G. E. Andrews (1986) q -Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, Vol. 66, Amer. Math. Soc., Providence, RI.
  • T. M. Apostol and I. Niven (1994) Number Theory. In The New Encyclopaedia Britannica, Vol. 25, pp. 14–37.
  • T. M. Apostol (1990) Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
  • T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.