# Dirichlet convolution of Möbius function with power of logarithm

In the article "Primes in tuples I" by Goldston, Pintz and Yıldırım, it is claimed that the Dirichlet convolution

$$$$\sum_{d\mid n}\mu(d)\left(\log \frac{n}{d}\right)^k$$$$ vanishes if $$n$$ has more than $$k$$ distinct prime factors. Here $$\log$$ denotes the natural logarithm. I have tried to show this, but I haven't been successful. I attempted a "brute force" solution by simply trying to evaluate the sum, but that was not successful. Clearly, the function isn't multiplicative either.

The Dirichlet series of this function is the $$k$$:th derivative of the reciprocal of the zeta function (Edit: This is wrong, see the accepted answer), but I don't see how that can help me. I would be grateful for any help.

• It is possibly to define arithmetic derivative for arithmetic function that obeys product rule. You may find synopsis in Apostol's book. Jul 1, 2021 at 7:07

Usually we define

$$$$\Lambda_k= \sum_{d\mid n}\mu(d)\left(\log \frac{n}{d}\right)^k$$$$

where $$\Lambda_0=\delta_0$$ and $$\Lambda_1$$ is the usual Von Mangoldt function.

Now the actual generating Dirichlet series for $$\Lambda_k$$ is given by $$(-1)^k\frac{\zeta^{(k)}}{\zeta}$$ as one can easily see since if $$L$$ is the logarithm operator, $$\Lambda_k=\mu*L^k$$, while in Dirichlet terms $$L$$ is the negative differential operator and $$L^k$$ is then $$(-1)^k$$ times the $$k$$th derivative of the function generated by $$1$$ which is $$\zeta$$

From this, it immediately follows that:

(noting that $$L\Lambda_k$$ is generated by $$-\frac{d}{ds}[(-1)^k\frac{\zeta^{(k)}}{\zeta}]$$ and that $$\Lambda*\Lambda_k$$ is generated by $$(-1)\frac{\zeta'}{\zeta} \times (-1)^k\frac{\zeta^{(k)}}{\zeta}$$)

$$\Lambda_{k+1}=L\Lambda_k+\Lambda*\Lambda_k$$ and now the result follows by induction:

Cases $$0,1$$ are clear, and since $$\Lambda$$ is supported on prime powers and $$\Lambda_k$$ on numbers with at most $$k$$ distinct prime factors by the inductive hypothesis, their convolution is supported on numbers with at most $$k+1$$ distinct prime factors pretty much by the definition of the convolution, while $$L\Lambda_k$$ is already supported only on numbers with at most $$k$$ distinct prime factors.

• Thank you, a very clear explanation. Jul 1, 2021 at 7:26
• happy to be of help Jul 1, 2021 at 15:46