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Totient summatory function
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Totient summatory function
In number theory, the totient summatory function is a summatory function of Euler's totient function defined by
It is the number of ordered pairs of coprime integers (p,q), where 1 ≤ p ≤ q ≤ n.
The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, ... (sequence A002088 in the OEIS). Values for powers of 10 are 1, 32, 3044, 304192, 30397486, 3039650754, ... (sequence A064018 in the OEIS).
Applying Möbius inversion to the totient function yields
where is the Möbius function. Φ(n) has the asymptotic expansion
where ζ(2) is the Riemann zeta function evaluated at 2, which is .
The summatory function of the reciprocal of the totient is
Edmund Landau showed in 1900 that this function has the asymptotic behavior
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Totient summatory function
In number theory, the totient summatory function is a summatory function of Euler's totient function defined by
It is the number of ordered pairs of coprime integers (p,q), where 1 ≤ p ≤ q ≤ n.
The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, ... (sequence A002088 in the OEIS). Values for powers of 10 are 1, 32, 3044, 304192, 30397486, 3039650754, ... (sequence A064018 in the OEIS).
Applying Möbius inversion to the totient function yields
where is the Möbius function. Φ(n) has the asymptotic expansion
where ζ(2) is the Riemann zeta function evaluated at 2, which is .
The summatory function of the reciprocal of the totient is
Edmund Landau showed in 1900 that this function has the asymptotic behavior