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 , where .
The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, ... . Values for powers of 10 are 1, 32, 3044, 304192, 30397486, 3039650754, ... .
Properties The following identity holds for all real : . This gives an implicit recurrence for the totient summatory function.
Applying Möbius inversion to the totient function or the above identity yields
where is the Möbius function. Then it can be shown that has the asymptotic expansion
where is the Riemann zeta function evaluated at 2, which is .
Reciprocal totient summatory function The summatory function of the reciprocal of the totient is
Edmund Landau showed in 1900 that this function has the asymptotic behavior
where is the Euler–Mascheroni constant,
and
The constant is sometimes known as Landau's totient constant. The sum converges to
In this case, the product over the primes in the right side is a constant known as the totient summatory constant, and its value is
See also Arithmetic function
References
External links OEIS Totient summatory function Decimal expansion of totient constant product(1 + 1/(p^2*(p-1))), p prime >= 2)