Ma–Chen–Wu asymptotic conjecture for truncated primes represented by the floor function

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Let 0<θ≤10<\theta\leq 1 be real, and let πθ(x)\pi_{\theta}(x) denote the number of integers nn with 1≤n≤xθ1\leq n\leq x^{\theta} such that ⌊x/n⌋\left\lfloor x/n\right\rfloor is prime. For a given integer L≥1L\geq 1, Ma–Chen–Wu's conjecture. For any 0<θ<10<\theta<1,

πθ(x)=∑k=1L(−1)k−1(k−1)!(1−θ)kxθ(log⁡x)k+O(xθ(log⁡x)L+1).\pi_{\theta}(x)=\sum_{k=1}^{L}\frac{(-1)^{k-1}(k-1)!}{(1-\theta)^k}\frac{x^{\theta}}{(\log x)^k}+O\left(\frac{x^{\theta}}{(\log x)^{L+1}}\right).

This conjecture extends the known asymptotic formula for πθ(x)\pi_{\theta}(x) in the range 23/47<θ<123/47<\theta<1 to every fixed 0<θ<10<\theta<1, with an arbitrarily long logarithmic expansion. The supplied text does not state whether the conjecture has been resolved.

References

Primary source

Runbo Li, “On some problems of primes with the floor function”, arXiv:2308.16301 (2023).

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