I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second Chebyshev function.
In Nagura's paper, he establishes the following inequality:
$$\psi\left(x\right) - \psi\left(\frac{x}{1806}\right) \le T\left(x\right) - T\left(\frac{x}{2}\right) - T\left(\frac{x}{3}\right) - T\left(\frac{x}{7}\right) - T\left(\frac{x}{43}\right) - T\left(\frac{x}{1806}\right) < 1.0851x $$
where $T\left(x\right) = \log\Gamma\left(\lfloor{x}\rfloor+1\right)$
Then, he establishes an upper bound using:
$$\psi\left(x\right) < 1.0851\left(x + \frac{x}{1806} + \frac{x}{1806^2} + \frac{x}{1806^3} + \ldots\right) < 1.086x$$
Using my analysis in this question and this question, and this effort to apply Stirling's formula in the same way as Nagura, then, for $x \ge 986$, I am finding:
$$T\left(\frac{x}{2}\right) - T\left(\frac{x}{3}\right) - T\left(\frac{x}{6}\right) < 0.321x$$
Applying the same approach as Nagura, I am finding:
$$T\left(\frac{x}{2}\right) - T\left(\frac{x}{3}\right) - T\left(\frac{x}{6}\right) \ge \psi\left(\frac{x}{2}\right) - \sum_{m=1}^{\infty}\left[\psi\left(\frac{x}{6m-3}\right) - \psi\left(\frac{x}{6m-2}\right) + \psi\left(\frac{x}{6m}\right) - \psi\left(\frac{x}{6m+2}\right) \right] \ge \psi\left(\frac{x}{2}\right) - \psi\left(\frac{x}{3}\right)$$
Putting it all together, I come up with:
$$\psi\left(\frac{x}{2}\right) < 0.321\left(x + \frac{x}{3} + \frac{x}{3^2} + \frac{x}{3^3} + \ldots\right) < \frac{3}{2}*0.321 = 0.4815x$$
Which, after seting $x = 2y$, results in:
$$\psi\left(y\right) < 0.4815\left(2y\right) = 0.963y$$
Considering the best known upper bound is $1.03883$, I am certainly doing something wrong.
Can anyone help me to figure out what's wrong with my analysis?
Thanks,
-Larry
Update: I wrote a simple java app to check $\psi(x)$. I am calculating $\psi(1627) > 1.01363*(1627)$. Hopefully, this will lead me to the mistake that I made. When I figure it out, I'll post it as part of this update.
So far, my suspicion is that the $0.321x$ is wrong. Based on $\psi(1627)$, it should be greater than $0.67575x$.
I found the mistake. :-)
This is wrong:
$$\psi\left(\frac{x}{2}\right) < 0.321\left(x + \frac{x}{3} + \frac{x}{3^2} + \frac{x}{3^3} + \ldots\right) < \frac{3}{2}*0.321 = 0.4815x$$
It should be:
$$\psi\left(\frac{x}{2}\right) < 0.321\left(x + \frac{2x}{3} + \frac{2^2{x}}{3^2} + \frac{2^3{x}}{3^3} + \ldots\right) < \frac{3}{1}*0.321 = 0.963x$$
