Totient summatory function explained

In number theory, the totient summatory function

\Phi(n) is a summatory function of Euler's totient function defined by:\Phi(n) : = \sum ^{n} _{k =1} \varphi(k), n\in N

It is the number of coprime integer pairs .

The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32 . Values for powers of 10 at .

Properties

Using Möbius inversion to the totient function, we obtain

\Phi(n) = \sum ^{n} _{k =1} k\sum _{d\mid} (\mu (d))/(d) = (1)/(2) \sum ^{n} ^{k =1} \mu(k) \left\lfloor (n)/(k) \right\rfloor \left( 1 + \left\lfloor (n)/(k) \right\rfloor \right)

has the asymptotic expansion

\Phi(n) \sim (1)/(2 \zeta( 2 )) n ^{2} + O\left( n log n \right ),

where is the Riemann zeta function for the value 2.

is the number of coprime integer pairs .

The summatory of reciprocal totient function

The summatory of reciprocal totient function is defined as

S(n) : = \sum { ^{n} _{k =1} (1)/(\varphi (k)) }

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

S(n) \sim A (\gamma +log n) + B + O\left( (log n)/(n\right))

where is the Euler–Mascheroni constant,

A = \sum ^{inf ty} _{k =1} (\mu (k) ^{2})/(k \varphi(k)) = (\zeta( 2 )\zeta( 3 ))/(\zeta( 6 )) = \prod _{p} \left( 1+ (1)/(p(p -1 )) \right)

and

B = \sum ^{inf ty} _{k =1} (\mu (k) ^{2log} k)/(k \varphi(k)) = A \prod _{p} \left( (log p)/(p ^{2-p+1}) \right).

The constant is sometimes known as Landau's totient constant. The sum

style \sum ^{inf ty (1)/(k\varphi (k))} ^{k =1} is convergent and equal to:\sum ^{inf ty} ^{k =1} (1)/(k\varphi (k)) = \zeta( 2 ) \prod _{p} \left( 1 + (1)/(p ^{2(p-1)}) \right) =2 . 20386 \ldots

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

\prod _{p} \left( 1+ (1)/(p ^{2(p-1)}) \right) = 1 . 339784 \ldots

See also

• Arithmetic function

External links

• OEIS Totient summatory function
• Decimal expansion of totient constant product(1 + 1/(p^2*(p-1))), p prime >= 2)