Extension of an arithmetic function like Gamma function

Question

As you know Gamma function is an extension of the factorial function. Is there any other function which is the extension of an arithmetic function? For example extension of the Euler's totient function?

Answer 1

As far as the prime counting function is concerned, we have the well-known asymptotics $\pi(x)\simeq$ $\dfrac x{\ln x}$ , which can indeed work as an ‘extension’, albeit the values are approximate, not exact. I am unaware of any meaningful closed-form formula that gives the exact values for this function. Also, for Euler's totient function we have $\varphi(x)\simeq\dfrac x{e^\gamma\ln\ln x}$ , with the same observations in place. But these probably resemble more Stirling's formula than an actual interpolation.