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{{Short description|Arithmetic function}}
In [[number theory]], the '''totient summatory function''' <math>\Phi(n)</math>  is a [[summatory function]] of [[Euler's totient function]] defined by

:<math>\Phi(n) := \sum_{k=1}^n \varphi(k), \quad n\in \mathbb{N}.</math>

It is the number of ordered pairs of [[Coprime integers|coprime]] integers {{Math|(''p'',''q'')}}, where {{Math|1 &le; ''p'' &le; ''q'' &le; ''n''}}.

The first few values are 0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, ... {{OEIS|A002088}}. Values for powers of 10 are 1, 32, 3044, 304192, 30397486, 3039650754, ... {{OEIS|A064018}}.

== Properties ==
Applying [[Möbius inversion formula|Möbius inversion]] to the totient function yields

:<math>\Phi(n) = \sum_{k=1}^n k\sum _{d\mid k} \frac {\mu (d)}{d} = \frac{1}{2} \sum _{k=1}^n \mu(k) \left\lfloor \frac {n}{k} \right\rfloor \left(1 + \left\lfloor \frac {n}{k} \right\rfloor \right).</math>

{{math|Φ(''n'')}} has the asymptotic expansion

:<math>\Phi(n) \sim \frac{1}{2\zeta(2)}n^{2}+O\left( n\log n \right ) = \frac{3}{\pi^2}n^2+O\left( n\log n \right),</math>

where {{math|ζ(2)}} is the [[Riemann zeta function]] evaluated at 2, which is <math>\frac{\pi^2}{6}</math>.<ref name="MWZETA2">{{mathworld|title=Riemann Zeta Function \zeta(2)|id=RiemannZetaFunctionZeta2|mode=cs2}}</ref>

== Reciprocal totient summatory function ==
The summatory function of the reciprocal of the totient is

:<math>S(n) := \sum _{k=1}^{n}{\frac {1}{\varphi (k)}}.</math>

[[Edmund Landau]] showed in 1900 that this function has the asymptotic behavior{{Citation needed|date=February 2025}}

:<math>S(n) \sim A (\gamma+\log n)+ B +O\left(\frac{\log n} n\right), </math>

where {{math|γ}} is the [[Euler–Mascheroni constant]],

:<math>A = \sum_{k=1}^\infty \frac{\mu (k)^2}{k \varphi(k)} = \frac{\zeta(2)\zeta(3)}{\zeta(6)} = \prod_{p\in\mathbb{P}} \left(1+\frac 1 {p(p-1)} \right),</math>

and

:<math>B = \sum_{k=1}^{\infty} \frac{\mu (k)^2\log k}{k \,\varphi(k)} = A \, \prod _{p\in\mathbb{P}}\left(\frac {\log p}{p^2-p+1}\right).</math>

The constant {{math|1= ''A''&nbsp;=&nbsp;1.943596...}} is sometimes known as '''Landau's totient constant'''. The sum <math>\textstyle \sum _{k=1}^\infty 1 / (k \; \varphi (k))</math> converges to

:<math>\sum _{k=1}^\infty \frac 1 {k\varphi (k)} = \zeta(2) \prod_{p\in\mathbb{P}} \left(1 + \frac 1 {p^2(p-1)}\right) =2.20386\ldots. </math>

In this case, the product over the primes in the right side is a constant known as the '''totient summatory constant''',<ref>{{OEIS2C|A065483}}</ref> and its value is

:<math>\prod_{p\in\mathbb{P}} \left(1+\frac 1 {p^2(p-1)} \right) = 1.339784\ldots.</math>

== See also ==
*[[Arithmetic function]]

== References ==
{{reflist}}
*{{MathWorld|id=TotientSummatoryFunction|title=Totient Summatory Function}}

==External links==
*[[oeis:wiki/Totient_summatory_function|OEIS Totient summatory function]]
* [https://oeis.org/A065483 Decimal expansion of totient constant product(1 + 1/(p^2*(p-1))), p prime >= 2)]

[[Category:Arithmetic functions]]


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