So I stumbled upon this while trying to prove something else and cannot find a simple (read:elementary) proof for this (I'm old fashioned and still use $\log$ for natural log):
For large enough even $x$, $\log(8x+1)+\log(8x+3)+\cdots+\log(9x-1)-\log 1-\log 3-\cdots-\log(x-1)>\vartheta(9x)-\vartheta(8x)$.
For general $x$ as $\sum_{\substack{8x<k\leq9x\\k\text{ odd}}}\log k-\sum_{\substack{k\leq x\\k\text{ odd}}}\log k>\vartheta(9x)-\vartheta(8x)$.
I know it holds for at least $x\geq 11000$ by applying bounds to the Chebyshev function and using Stirling's approximation, but I would like to avoid using said Chebyshev bounds as they were proven with analytic results and I'm hoping for a more elementary approach.
Thanks in advance.