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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 $X$?
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    $\begingroup$ 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. $\endgroup$
    – Ofir Gorodetsky
    Commented Dec 16, 2015 at 18:08

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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).

This sort of statement, for other functions, goes back to Littlewood's oscillation theorems and continues today in the subfield of comparative prime number theory. See "Limiting distributions of the classical error terms of prime number theory" by Akbary, Ng, and Shahabi for a general framework that addresses problems such as this.

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  • $\begingroup$ Great! But what can be proved, if anything, unconditionally? $\endgroup$
    – Jeffrey Shallit
    Commented Dec 16, 2015 at 23:04
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    $\begingroup$ 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. $\endgroup$
    – Peter Humphries
    Commented Dec 17, 2015 at 2:36
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    $\begingroup$ 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 $\psi(x)$. $\endgroup$
    – Lucia
    Commented Dec 17, 2015 at 2:51
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    $\begingroup$ To get the (Z_{gamma}) independent, one probably also needs to assume linear independance of ordinates of zeros of zeta... $\endgroup$
    – Denis Chaperon de Lauzières
    Commented Dec 17, 2015 at 7:16

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