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1: 27.8 Dirichlet Characters
27.8.6 r = 1 ϕ ( k ) χ r ( m ) χ ¯ r ( n ) = { ϕ ( k ) , m n ( mod k ) , 0 , otherwise .
A Dirichlet character χ ( mod k ) is called primitive (mod k ) if for every proper divisor d of k (that is, a divisor d < k ), there exists an integer a 1 ( mod d ) , with ( a , k ) = 1 and χ ( a ) 1 . If k is prime, then every nonprincipal character χ ( mod k ) is primitive. …
2: 27.2 Functions
27.2.1 n = r = 1 ν ( n ) p r a r ,
27.2.8 a ϕ ( n ) 1 ( mod n ) ,
and if ϕ ( n ) is the smallest positive integer f such that a f 1 ( mod n ) , then a is a primitive root mod n . …
27.2.12 μ ( n ) = { 1 , n = 1 , ( - 1 ) ν ( n ) , a 1 = a 2 = = a ν ( n ) = 1 , 0 , otherwise .
27.2.13 λ ( n ) = { 1 , n = 1 , ( - 1 ) a 1 + + a ν ( n ) , n > 1 .
3: 27.10 Periodic Number-Theoretic Functions
This is the sum of the n th powers of the primitive k th roots of unity. … For a primitive character χ ( mod k ) , G ( n , χ ) is separable for every n , and
27.10.11 | G ( 1 , χ ) | 2 = k .
Conversely, if G ( n , χ ) is separable for every n , then χ is primitive (mod k ). The finite Fourier expansion of a primitive Dirichlet character χ ( mod k ) has the form …
4: 27.21 Tables
8 gives examples of primitive roots of all primes 9973 ; Table 24. …
5: 29.17 Other Solutions
They are algebraic functions of sn ( z , k ) , cn ( z , k ) , and dn ( z , k ) , and have primitive period 8 K . …
6: 25.15 Dirichlet L -functions
where χ 0 is a primitive character (mod d ) for some positive divisor d of k 27.8). When χ is a primitive character (mod k ) the L -functions satisfy the functional equation: … Since L ( s , χ ) 0 if s > 1 , (25.15.5) shows that for a primitive character χ the only zeros of L ( s , χ ) for s < 0 (the so-called trivial zeros) are as follows: …
7: 24.16 Generalizations
§24.16(ii) Character Analogs
Let χ be a primitive Dirichlet character mod f (see §27.8). …