Above is the definition i got from my note.
I was trying to do these and i get stuck when i complete setting up the definition. I am trying to break down $c(n) = ((e_1 - 2e_2) * u)n$
= $( u * e_1 - u * 2e_2)(n)$
Are there any way that i can keep doing that question?
And for the second bit,
I have tried but i can't really figure out how does the $ \frac{1}{1-2^{(1－s)}}$ come out
anyone can give me some help?
Thanks
A more conventional notation for those $e_j$ would be $\delta_{j}$ - the Kronecker delta.
Now, those convolutions. What does convolution with $e_j$ do? $$(e_j*p)(n) =\sum_{d|n}e_j(d)\cdot p(n/d) = \begin{cases}p(n/j)& j|n\\0&\text{otherwise}\end{cases}$$ We space out the entries so they're at distance $j$ instead of $1$, filling in between them with zeros. Note that $e_1$ acts as the identity here; $e_1*p = p$ for all $p$.
What about the Dirichlet series> Well, if $f(s) = \sum_n p(n)n^{-s}$ and $g(s)=\sum_n(e_j*p)(n)n^{-s}$, we get $$g(s) = \sum_n(e_j*p)(n)n^{-s} = \sum_{j|n} p(n/j)n^{-s} = \sum_k p(k)(jk)^{-s}=j^{-s}\sum_k p(k)k^{-s}= j^{-s}f(s)$$
So then, the specific problem. You have a particular $u$ - which you still haven't defined, but I know what it is from context - so then we get $$(e_1*u)(n)-2(e_2*u)(n) = \begin{cases}u(n)-2u(n/2)& n\text{ is even}\\ u(n)& n\text{ is odd}\end{cases}$$ Can you evaluate this? Can you then take the series and get the formula relating the zeta function and the series for these coefficients?