# nt.number theory – Any more improvement for \$|pi-22dfrac{4times7^{2n+1}-4times7^{2n-3}+7^{2n-4}}{4times7^{2n+2}}|

This is a part of my research about irrationality measure of $$pi$$ and sum of power divisor function $$sigma$$, I have used the following ratio of odd iteration and even iteration of sum power divisor function $$dfrac{sigma_{2n+1}(p)}{sigma_{2n+2}(p)}$$ for $$p=7$$ which the ratio close to $$1/7$$. Multiply $$dfrac{sigma_{2n+1}(p)}{sigma_{2n+2}(p)}$$ by $$22$$ the value of that sequences would be close to $$22/7$$. It is known that $$sigma_n(p)=n^p+1$$ with $$p$$ is a prime number. Just a small improvement I come up to the following integer sequences $$a_n$$, and $$b_n$$ such that :
$$a_n=22*(4times7^{2n+1}-4times^{2n-3}+7^{2n-4}) quadtext{and}quad b_n=4times7^{2n+2}, quad n>0$$
such that $$a_n/b_n to pi$$ with approximation of $$10 ^{-6}$$. Now my research is the determination of the values of both $$C$$ and $$delta$$ to get a bound for the difference in the below absolute value or LHS of the following inequality:
$$begin{equation} leftlvertpi-22dfrac{4times7^{2n+1}-4times7^{2n-3}+7^{2n-4}}{4times7^{2n+2}}rightrvert

With the choice of $$C=e^{e^{e^{e^{e}}}}$$ the inequality holds up to $$10^{10}$$ using mathematica code with $$delta$$ in the range $$(0,1)$$. Now I have two question regarding the prediction of irrationality measure of $$pi$$ rathar than that the upper bound of LHS of inequality eqref{1}:

Question

1. Must $$delta$$ always lie in $$(0,1)$$ with the choice of large real number $$C$$ and with the choice of integer sequences $$a_n$$ and $$b_n$$?
2. Is it possible to find integer polynomial $$P_n$$ to make the ratio $$dfrac{a_n+p_n}{b_n+p_n}$$ as close as possible to $$pi$$ which lead to have a stronger bound which it self lead to predict the exact irrationality measure of $$pi$$?