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By Landau's theorem on Dirichlet series, we know that all the step functions ($k\geq 1$)

$$M_k(x)=\frac{1}{2\pi i}\int^{2+i\infty}_{2-i\infty}\frac{x^sds}{\zeta^k(s)s}=\sum_{n\leq x}\prod_{p|n}{\alpha_p-k-1\choose\alpha_p},$$ change sign for indefinitely large values of $x$ (here $n=\prod_{p|n}p^{\alpha_p}$ is the prime factorisation).

Since $M_k$ is a non-constant polynomial in $k$ of degree $\leq\log_2x$, certainly no more than $\log_2 x$ of the functions $M_k$ may vanish simultaneously for a given value of $x$. I would like to know anything that can be said about the following:

  1. Do there exist values of $x$ for which $$M_1(x)=M_2(x)=\cdots=M_{\log_2x-2}(x)=0?$$

  2. What hypotheses preclude the simultaneous vanishing of $M_k(x)$, $1\leq k\leq \log_2 x-2$?

  3. Is it possible that $M_k(x)$, $1\leq k\leq \log_2 x-2$, can be simultaneously small?

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  • $\begingroup$ I can see that Landau's theorem could be used to deduce $M_k$ changes sign infinitely often, but how could it be used to force it to be zero infinitely often? $\endgroup$
    – Terry Tao
    Nov 2, 2014 at 23:58
  • $\begingroup$ Thanks, I can't see that it does actually. I have edited the question. $\endgroup$ Nov 3, 2014 at 7:10

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