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A quantitative refinement of Lu's elementary prime number theorem

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Rajat Bhattacharjee ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We refine Lu's elementary method for the prime number theorem to obtain, for every fixed K>400K>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 ψ\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 terms. The resulting remainder strengthens Lu's fixed-parameter estimates and Goz\'e's quantitative refinement within the real-variable framework; stronger remainder estimates are known by other methods.

References

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Version history

  1. v1Submitted by Rajat BhattacharjeeInitial depositCurrentOct 21, 2026