# number theory

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##### 1: 27.18 Methods of Computation: Primes
An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). …
##### 2: 27.1 Special Notation
###### §27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). …
##### 4: 27.17 Other Applications
There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …
For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … If an odd integer $P$ has prime factorization $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}$, with $(n|1)=1$. …
##### 6: 20.12 Mathematical Applications
###### §20.12(i) NumberTheory
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.