A quantitative refinement of Lu's elementary prime number theorem
We refine Lu's elementary method for the prime number theorem to obtain, for every fixed $K>400$, \[ \psi(X)-X=O_K\!\left( X\exp\!\left\{-\frac{\sqrt{\log X}}{(\log\log X)^K}\right\}\right) \qquad(X\to\infty), \] where $\psi$ is Chebyshev's function. The main ingredient is a comparison estimate that remains uniform as the degree grows and the short-interval parameter shrinks. Combining this estimate with exponential weights in Balog's identity, a single finite energy step and positivity-based interpolation yields the stated bound, retaining the full degree tail and all arithmetic correction…
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