primitive%20roots
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1: 27.2 Functions
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27.2.1
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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27.2.8
►and if is the smallest positive integer such that , then is a primitive root mod .
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27.2.12
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2: 27.10 Periodic Number-Theoretic Functions
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►This is the sum of the th powers of the primitive
th roots of unity.
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►For a primitive character , is separable for every , and
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27.10.11
►Conversely, if is separable for every , then is primitive (mod ).
►The finite Fourier expansion of a primitive Dirichlet character has the form
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3: 27.21 Tables
4: 27.8 Dirichlet Characters
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27.8.6
►A Dirichlet character is called primitive (mod ) if for every proper divisor of (that is, a divisor ), there exists an integer , with and .
If is prime, then every nonprincipal character is primitive.
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5: 1.11 Zeros of Polynomials
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►Roots of are , , .
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►The square roots are chosen so that
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§1.11(iv) Roots of Unity and of Other Constants
►The roots of … ►The roots of …6: 29.17 Other Solutions
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►They are algebraic functions of , , and , and have primitive period .
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7: 20 Theta Functions
Chapter 20 Theta Functions
…8: 25.15 Dirichlet -functions
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►where is a primitive character (mod ) for some positive divisor of (§27.8).
►When is a primitive character (mod ) the -functions satisfy the functional equation:
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►Since if , (25.15.5) shows that for a primitive character the only zeros of for (the so-called trivial zeros) are as follows:
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9: 36.5 Stokes Sets
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►where are the two smallest positive roots of the equation
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►For the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation
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►Here is the root of the equation
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►where is the root of the equation
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►where is the positive root of the equation
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