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

From papers

Let 0<θ10<\theta\leq 1 be real, and let πθ(x)\pi_{\theta}(x) denote the number of integers nn with 1nxθ1\leq n\leq x^{\theta} such that x/n\left\lfloor x/n\right\rfloor is prime. For a given integer L1L\geq 1, Ma–Chen–Wu's conjecture. For any 0<θ<10<\theta<1,

πθ(x)=k=1L(1)k1(k1)!(1θ)kxθ(logx)k+O(xθ(logx)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.

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Sources & referencesView supporting material

Primary source

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

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