# Questions tagged [dirichlet-convolution]

Use this tag for questions related to Dirichlet convolution in number theory

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### Dirichlet convolution inverse of Euler's totient function

Let g(n) be the convolution inverse of Euler's totient function $\varphi(n)$. Let $n=p_1^{a_1}...p_t^{a_t}$, where $p_j$ are the distinct prime divisors of $n$. Find a formula for $g(n)$ and prove ...
1 vote
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### Book reference for studying Dirichlet Convolution

Now I am studying elementary number theory, I am interested in arithmetic function, I have studied Burton's Number Theory but I can't find Dirichlet Convolution as a particular topic, I will be highly ...
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### Dirichlet convolution: $f \ast Id = 1$

How do I find a $f$ such that $f \ast Id = 1$ where $Id$ denotes the identity function, $f(x)=x$ , and $1(x)=1$? I tried convolving both sides with $\mu$ and got $f \ast \phi = \epsilon$ which gave no ...
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### Discrete Fourier Transform & GCD

While reading Wolfgang Schramm's original paper concerning the relationship between the discrete Fourier transform and gcd, I came across the following condensed argument for his more general theorem ...
1 vote
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I am using the example from Theorem 1 of Friedrich Pillichshammer "Euler's Constant and Averages of Fractional Parts" (https://www.dmg.tuwien.ac.at/nfn/gamma.pdf) we have for integers $k >... 2 votes 2 answers 183 views ### Is it possible turn the Dirichlet ring into a Banach algebra? The set of all arithmetic functions$f:\mathbb{Z}^{+}\to\mathbb{C}$, under pointwise addition and Dirichlet convolution, is a commutative ring, not all functions are Dirichlet invertible. So my ... 0 votes 1 answer 300 views ### Dirichlet convolution of the small prime omega function and the Mobius function I have seen that: $$(\omega\star\mu)(n)=\sum_{d\vert n}\mu(d)\omega\left(\frac{n}{d}\right)=\begin{cases}1 & n\ \text{is prime}\\ 0 &\text{otherwise} \end{cases}$$ where$\mu(n)=\delta_{\omega(... 30 views

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### Question on the coefficient of the Dirichlet series related to $\frac{\zeta(s+2)}{\zeta(s)}$
This question is about the evaluation of $a(n)$ defined in (1) below which is related to the Riemann zeta function $\zeta(s)$ as illustrated in (2) below. (1) \$\quad a(n)=\sum\limits_{d|n}\frac{\mu\...