Are the coefficients of li(x)’s asymptotic expansion optimal among approximants of the form x.P(1/log x)?
问题内容
- States the class — x·P(1/L), L = log x — and li's expansion x Σ (k−1)!/L^k. 2. Question 1: is it standard that those coefficients are the unique optimum, and is there a canonical reference? 3. Shows why you're asking: your π_g, your π_h^(N), the observation that x^{1/n} for n≥2 is O(√x) and drops out of every coefficient, the expansion (u+u²)/(1+u²) = u+u²−u³−u⁴+…, the resulting vectors (1,1,2,1,0,…) for N≥4, and the conclusion π_h^(N) − π ~ −5x/(log x)⁴ independent of N — with the numerical check (5.19 at 10⁶⁰, 5.01 at 10²⁰⁰⁰). 4. Question 2: among rational R(L) pole-free for x>1, how many li-coefficients can be matched? Padé to Σ j!uʲ has poles on u>0 (Stieltjes cut), so n≥1 puts a pole in x>1, while [m/0] is pole-free and matches m+1. Is that the whole picture, and is there a reference for the pole locations?
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