# What is known about $\sum_{n \leq x} \mu(n) \varphi(n)$?

Let $\mu(n)$ denote the Möbius function and $\varphi(n)$ the Euler-phi function. What is known about $f(x) = \sum_{n \leq x} \mu(n) \varphi(n)$? For example:

1. Is it known that $f(x)$ grows without bound?
2. Is it known that $-f(x)$ grows without bound?
3. Is it known that $f(x)$ crosses the origin infinitely often?
4. Are there are any solutions to $f(x) = 0$ for $x \geq 3$? (I didn't find any for $x < 10^8$.)
5. What are some good upper and lower bounds for $|f(x)|$?
6. How many times does $f$ cross the origin for $x < X$, as a function of $X$?
• The Dirichlet series of $\mu(n)\phi(n)$ is of the form $F(s)/\zeta(s-1)$, where $F$ is analytic in $s\ge 2$. This might be of use, maybe some Tauberian theorem can settle some of your questions. – Ofir Gorodetsky Dec 16 '15 at 18:08

Fleshing out Ofir Gorodetsky's comment: if we define $G(s) = \sum_{n=1}^\infty \mu(n)\phi(n)n^{-s}$, then we have $G(s) = F(s)/\zeta(s-1)$ where $$F(s) = \prod_p \bigg( 1 - \frac1{p^s-p} \bigg)$$ is absolutely convergent for $\Re s>1$. The rightmost singularities of $G(s)$ are therefore at the points $1+\rho$ where $\rho$ denotes nontrivial zeros of $\zeta(s)$. Assuming the Riemann hypothesis, we thus expect $f(x)/x^{3/2} = x^{-3/2} \sum_{n\le x} \mu(n)\phi(n)$ to have a limiting logarithmic distribution, which will be the same as the distribution of the random variable $$\sum_{\gamma} \bigg |\frac{F(3/2+i\gamma)}{(3/2+i\gamma)\zeta'(1/2+i\gamma)}\bigg| \Re Z_\gamma,$$ where $\gamma$ runs over the positive imaginary parts of zeros of $\zeta(s)$, and $\{Z_\gamma\}$ is an independent set of random variables each uniformly distributed on the unit circle in $\Bbb C$. (The independence of the $\{Z_\gamma\}$ is based on our belief that the imaginary parts of the nontrivial zeros of $\zeta(s)$ are linearly independent over the rationals.)
This distribution will be roughly bell-shaped but not normal, indeed probably decaying faster than a normal distribution. The function $f(e^y)/e^{3y/2}$ will be an almost periodic function with mean $0$, but will presumably be unbounded above and unbounded below. Numerical computations up to $x=10^6$ support these claims (other than the unboundedness, which will be extremely gradually realized).
• See chapter 15 of Montgomery and Vaughan's book; the methods there produce sign changes for $\psi(x)$, and the same methods would show that $f(x)/x^{3/2}$ changes sign infinitely often. – Peter Humphries Dec 17 '15 at 2:36
• You might need to change your explicit formula a bit -- there'll be $1/\zeta^{\prime}(\rho)$'s in the denominator. So like with the usual partial sums of the Mobius function, this would be a bit more painful to work with than the analogs for $\psi(x)$. – Lucia Dec 17 '15 at 2:51