I am interested in partial sums $$T(x) = \sum_{n \le x} \tau(n)$$
Where $\tau$ is Ramanujan's Tau function.
It is known that $|\tau(n)| = O( n^{5.5+\epsilon})$ for $\epsilon>0$ and that this is essentially the best bound. Hence a trivial bound on $T(x)$ is $O( x^{6.5+\epsilon})$. But $\tau$ oscillates its sign, so there seems to exist further cancellations.
Indeed, I did some calculations in my computer and I found that for $x$ up to $10^6$, $|T(x)|$ is never greater than $x^{5.73}$.
Are there any better upper bounds for T(x)?
I couldn't find any sources.
I am interested mostly because of the implications for the "Ramanujan L-function":
$$L(\Delta,s) = \sum_{n=1}^{\infty} \frac{\tau(n)}{n^s}$$
The upper bound $\tau(n) = O(n^{5.5+\epsilon})$ is only enough to define this function for $\Re(s) > 6.5$, but by partial summation:
$$L(\Delta,s) = \sum_{n=1}^{\infty} T(n)(\frac{1}{n^s}-\frac{1}{(n+1)^s})$$
Hence an upper bound for $T(x)$ of the kind $O(x^r)$ would be enough to extend this function to $\Re(s)>r$. So upper bounds better than $x^{6.5}$ for $T(x)$ will further extend this function.
If the bound holds for any $r<6$, that would prove $L(\Delta,s)$ is holomorphic on $\Re(s)>r$, which together with the symmetry that this function has about the strip $\sigma = 6$ would be enough to extend it for the whole plane and prove it is entire. I was also not able to confirm whether $L(\Delta,s)$ is entire or not, so I would also appreciate sources for that.