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Totient summatory function

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In number theory, the totient summatory function is a summatory function of Euler's totient function defined by:

It is the number of coprime integer pairs {p, q}, 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 at (sequence A064018 in the OEIS).

Properties

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Using Möbius inversion to the totient function, we obtain

Φ(n) has the asymptotic expansion

where ζ(2) is the Riemann zeta function for the value 2.

Φ(n) is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n.

The summatory of reciprocal totient function

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The summatory of reciprocal totient function is defined as

Edmund Landau showed in 1900 that this function has the asymptotic behavior

where γ is the Euler–Mascheroni constant,

and

The constant A = 1.943596... is sometimes known as Landau's totient constant. The sum is convergent and equal to:

In this case, the product over the primes in the right side is a constant known as totient summatory constant,[1] and its value is:

See also

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References

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  • Weisstein, Eric W. "Totient Summatory Function". MathWorld.
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