# multiplicative number theory

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## 1—10 of 21 matching pages

##### 1: 27.1 Special Notation
###### §27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). …
##### 3: 27.4 Euler Products and Dirichlet Series
27.4.1 $\sum_{n=1}^{\infty}f(n)=\prod_{p}\left(1+\sum_{r=1}^{\infty}f(p^{r})\right),$
Euler products are used to find series that generate many functions of multiplicative number theory. …
##### 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 $\left(n,k\right)=1$, then the characters satisfy the orthogonality relationA divisor $d$ of $k$ is called an induced modulus for $\chi$ 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.