Euler%20totient
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1: 24.1 Special Notation
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Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 27.2 Functions
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►This is the number of positive integers that are relatively prime to ; is Euler’s totient.
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►and if is the smallest positive integer such that , then is a primitive root mod .
The numbers are relatively prime to and distinct (mod ).
…Note that .
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►Table 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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3: 27.16 Cryptography
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►Thus, and .
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►To do this, let denote the reciprocal of modulo , so that for some integer .
(Here is Euler’s totient (§27.2).)
By the Euler–Fermat theorem (27.2.8), ; hence .
But , so is the same as modulo .
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4: 27.3 Multiplicative Properties
5: 27.6 Divisor Sums
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►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors.
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27.6.5
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27.6.6
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6: 27.21 Tables
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►Glaisher (1940) contains four tables: Table I tabulates, for all : (a) the canonical factorization of into powers of primes; (b) the Euler totient
; (c) the divisor function ; (d) the sum of these divisors.
Table II lists all solutions of the equation for all , where is defined by (27.14.2).
…6 lists , and for ; Table 24.
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7: 27.8 Dirichlet Characters
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►For any character , if and only if , in which case the Euler–Fermat theorem (27.2.8) implies .
There are exactly different characters (mod ), which can be labeled as .
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27.8.6
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8: 27.11 Asymptotic Formulas: Partial Sums
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►where is Euler’s constant (§5.2(ii)).
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►where again is Euler’s constant.
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27.11.6
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27.11.7
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►The prime number theorem for
arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if , then the number of primes with is asymptotic to as .
9: 27.5 Inversion Formulas
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27.5.4
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