[latexpage]
It seems like you are exploring the relationship between ( f(u) ) and the non-trivial zeros of the Riemann zeta function, specifically focusing on how the residual term ( \text{Residual}_T(u) ) in your function behaves as ( T ) approaches infinity, particularly in the context of worst-case scenarios for ( u ).
The function ( f(u) = -\sum_{\rho} \frac{\exp(i \cdot \text{Im}(\rho) \cdot u)}{\rho} ), where the sum is over the non-trivial zeros ( \rho ) of the Riemann zeta function, is an interesting way to explore the relationship between these zeros and the prime counting function, especially in the context of the explicit formula for the Chebyshev function.
As ( T ) becomes very large, the sum ( S_T(u) ), which includes terms corresponding to zeros up to a certain height ( T ), will capture more of the high-frequency oscillations in the prime counting function. The residual term ( \text{Residual}_T(u) ), which includes the contributions from zeros above height ( T ), would ideally decrease as ( T ) increases, because the main contribution to the sum comes from the zeros up to height ( T ).
However, the behavior of ( \text{Residual}_T(u) ) as ( T ) approaches infinity can be complex and depends on the distribution and properties of the high zeros of the zeta function. In the worst-case scenarios for ( u ), where ( u ) corresponds to points where the prime counting function shows significant deviation from its expected behavior (like around the Skewes number), ( \text{Residual}_T(u) ) could still contribute significantly, even for large ( T ).
This is due to the fact that the terms in the sum become increasingly oscillatory and their amplitudes diminish more slowly as you consider zeros higher up the critical line. Hence, the impact of high-height zeros can’t be completely disregarded, especially when exploring the behavior of ( f(u) ) in regions where prime number theory suggests anomalies or significant deviations.
Your approach indicates a deep exploration of these relationships, potentially offering insights into the complex interplay between the zeros of the zeta function and the distribution of prime numbers. This kind of analysis is at the forefront of understanding the mysteries of prime numbers and their connection to the zeros of the Riemann zeta function.
