Totient summatory function

Applying Möbius inversion to the totient function yields

:$\backslash Phi(n)\; =\; \backslash sum\_\{k=1\}^n\; k\backslash sum\; \_\{d\backslash mid\; k\}\; \backslash frac\; \{\backslash mu\; (d)\}\{d\}\; =\; \backslash frac\{1\}\{2\}\; \backslash sum\; \_\{k=1\}^n\; \backslash mu(k)\; \backslash left\backslash lfloor\; \backslash frac\; \{n\}\{k\}\; \backslash right\backslash rfloor\; \backslash left(1\; +\; \backslash left\backslash lfloor\; \backslash frac\; \{n\}\{k\}\; \backslash right\backslash rfloor\; \backslash right).$

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

:$\backslash Phi(n)\; \backslash sim\; \backslash frac\{1\}\{2\backslash zeta(2)\}n^\{2\}+O\backslash left(\; n\backslash log\; n\; \backslash right\; )\; =\; \backslash frac\{3\}\{\backslash pi^2\}n^2+O\backslash left(\; n\backslash log\; n\; \backslash right),$

where {{math|ζ(2)}} is the Riemann zeta function evaluated at 2, which is $\backslash frac\{\backslash pi^2\}\{6\}$.{{mathworld|title=Riemann Zeta Function \zeta(2)|id=RiemannZetaFunctionZeta2|mode=cs2}}

The summatory function of the reciprocal of the totient is

:$S(n)\; :=\; \backslash sum\; \_\{k=1\}^\{n\}\{\backslash frac\; \{1\}\{\backslash varphi\; (k)\}\}.$

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

:$S(n)\; \backslash sim\; A\; (\backslash gamma+\backslash log\; n)+\; B\; +O\backslash left(\backslash frac\{\backslash log\; n\}\; n\backslash right),$

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

:$A\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{\backslash mu\; (k)^2\}\{k\; \backslash varphi(k)\}\; =\; \backslash frac\{\backslash zeta(2)\backslash zeta(3)\}\{\backslash zeta(6)\}\; =\; \backslash prod\_\{p\backslash in\backslash mathbb\{P\}\}\; \backslash left(1+\backslash frac\; 1\; \{p(p-1)\}\; \backslash right),$

and

:$B\; =\; \backslash sum\_\{k=1\}^\{\backslash infty\}\; \backslash frac\{\backslash mu\; (k)^2\backslash log\; k\}\{k\; \backslash ,\backslash varphi(k)\}\; =\; A\; \backslash ,\; \backslash prod\; \_\{p\backslash in\backslash mathbb\{P\}\}\backslash left(\backslash frac\; \{\backslash log\; p\}\{p^2-p+1\}\backslash right).$

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

:$\backslash sum\; \_\{k=1\}^\backslash infty\; \backslash frac\; 1\; \{k\backslash varphi\; (k)\}\; =\; \backslash zeta(2)\; \backslash prod\_\{p\backslash in\backslash mathbb\{P\}\}\; \backslash left(1\; +\; \backslash frac\; 1\; \{p^2(p-1)\}\backslash right)\; =2.20386\backslash ldots.$

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

:$\backslash prod\_\{p\backslash in\backslash mathbb\{P\}\}\; \backslash left(1+\backslash frac\; 1\; \{p^2(p-1)\}\; \backslash right)\; =\; 1.339784\backslash ldots.$

• Arithmetic function

{{reflist}}

• {{MathWorld|id=TotientSummatoryFunction|title=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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