Definition
A function ƒ(k) on positive integers is d-Ramanujan of order α if there exists a polynomial p and a positive constant c such that
![{\displaystyle \displaystyle |f(k)-(d-1)^{k}p(k)|\leq ck^{c}a^{k}\quad \forall k\geq 1}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/787af3c4d52455b394d9bd878f05d288e22812af)
We call (d-1)kp(k) the principal term of the function and ƒ(k)-(d-1)kp(k) its error term.
A d-Ramanujan function of order √d-1 is simply called a d-Ramanujan function.
Properties
Ramanujan functions are closed under addition and convolution. More precisely, let ƒ1 and ƒ2 be two d-Ramanujan functions of order α then
- ƒ1 + ƒ2 is d-Ramanujan of order α
Proof
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Assume that
![{\displaystyle {\begin{aligned}f_{1}(k)&=(d-1)^{k}p_{1}(k)+e_{1}(k),\quad |e_{1}(k)|\leq c_{1}k^{c_{1}}\alpha ^{k}\\f_{2}(k)&=(d-1)^{k}p_{2}(k)+e_{2}(k),\quad |e_{2}(k)|\leq c_{2}k^{c_{2}}\alpha ^{k}\quad \forall k\geq 1\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/98fae3ab4b39e17ce8b1d67c61286a65a0ba9049)
then clearly
![{\displaystyle {\begin{aligned}|(f_{1}+f_{2})(k)-(d-1)^{k}(p_{1}+p_{2})(k)|&=|e_{1}(k)+e_{2}(k)|\\&\leq |e_{1}(k)|+|e_{2}(k)|\\&\leq 2ck^{c}\alpha ^{k}\quad \forall k\geq 1\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/282451c8066031ac23cf67c1bba10d203107b4f4)
where c is the largest of the constants c1 and c2.
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- ƒ1 * ƒ2 is d-Ramanujan of order α
Proof
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First, recall the definition of the convolution of two functions:
![{\displaystyle \displaystyle (f_{1}*f_{2})(k)=\sum _{j=1}^{k-1}f_{1}(j)f_{2}(k-j)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/b1ec9c6d4ae402487e9b9dc839053bb1b52a00bc)
Assuming that
![{\displaystyle {\begin{aligned}f_{1}(k)&=(d-1)^{k}p_{1}(k)+e_{1}(k),\quad |e_{1}(k)|\leq c_{1}k^{c_{1}}\alpha ^{k}\\f_{2}(k)&=(d-1)^{k}p_{2}(k)+e_{2}(k),\quad |e_{2}(k)|\leq c_{2}k^{c_{2}}\alpha ^{k}\quad \forall k\geq 1\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/98fae3ab4b39e17ce8b1d67c61286a65a0ba9049)
TO CONTINUE
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To do
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Continue the above proof
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