Expansion of inverse logarithmic integral in terms of lambert w

Question

Cipolla and Césaro both gave expansions of $\operatorname{li}^{-1}$ in tems of nested $\log$ functions. I think it can be written in terms of the Lambert-W function in the form:

$\operatorname{li}^{-1}(n)=n\sum _{i=1} a_i(-1)^{i+1} W_{-1}\left(-e/n\right){}^{2-i} $

where $a_i$ begins: $\small{-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, \dots}$

where the next one $\small{\approx 1465235 + \epsilon, \ \epsilon =-1/12?)}$.

The above should be accurate to $o(1/\log(n)^{10})$ - is this correct? Is there a nice way to obtain the coefficients other than by computation?

For convenience:

f[n_] := With[
{a = {-1, 0, 1, 3, 11, 105/2, 613/2, 12635/6, 99677/6, 1774391/12, 17582819/12}}, 
n Sum[(-1)^(1 + k) a[[k]] ProductLog[-1, -E/n]^(2 - k), {k, 1, 11}]];
g[n_] := Quiet[x /. FindRoot[LogIntegral@x == n, {x, N[n Log[n], 200]}, 
WorkingPrecision -> 200]];

Abs@Log@N[1/Log[#]^10] &[10^10^7]
Abs@Log@N[1 - f@#/g@#] &[10^10^7]

Answer 1

Let me elaborate on reuns' idea and show how one can find coefficients $a_i$ by solving a certain ODE.

Let's define: $$f(z) := \sum_{i\geq 3} a_i z^{i-3}$$ so that we get a functional equation: $$\mathrm{li}\big( -x(\frac{1}{t}+tf(t)) \big) = x,$$ where $t=t(x):=-\frac{1}{W_1(-e/x)}$. Differentiating this equation with respect to $x$, and then substituting $x=\frac{et}{\exp(-1/t)}$, we obtain a differential equation: $$(\star)\qquad t^3f'(t) - t(1-2t)f(t) - (1-t)\log(1-t^2f(t)) + t = 0$$ with the initial condition $f(0)=1$.

I've played with ODE $(\star)$ in Maple. In principle, Maple can solve it in the following form: $$f(t) = \frac{1-\exp\big(r-1/t\big)}{t^2},$$ where $r$ is the root of the equation: $\mathrm{Ei}(1,-1-r)=-\frac{et}{\exp(-1/t)}$. However, I'm not sure if this implicit form is that useful.

On the other hand, Maple can solve $(\star)$ in power series of given order, thus computing the coefficients $a_i$. For example,

Order:=15: dsolve( { t^3*diff(f(t),t) - t*(1-2*t)*f(t) - (1-t)*log(1-t^2*f(t)) + t = 0, f(0)=1 }, f(t), series);

gives

$$f \left( t \right) =1+3\,t+11\,{t}^{2}+{\frac{105}{2}}{t}^{3}+{\frac{613}{2}}{t}^{4}+{\frac{12635}{6}}{t}^{5}+{\frac{99677}{6}}{t}^{6}+{\frac{1774391}{12}}{t}^{7}+{\frac{17582819}{12}}{t}^{8}+{\frac{1919343719}{120}}{t}^{9}+{\frac{22882040099}{120}}{t}^{10}+{\frac {295793507053}{120}}{t}^{11}+{\frac{1373607474819}{40}}{t}^{12}+{\frac{323119030735871}{630}}{t}^{13}+{\frac{20600974525589671}{2520}}{t}^{14}+O \left( {t}^{15} \right).$$

I've added the sequences of coefficients numerators / denominators to the OEIS as A337734 and A337735, respectively.

Answer 2

You may find the following result interesting. The point is to highlight that by reasoning in terms of $p_n -n $ it is possible to find new expressions of the asymptotic trend.

let $p_{n}$ be the nth-prime number and $W_{t}$ the Lambert-W function / $W(z)e^{W(z)}=z $ /:

$ \exists \ \ l \in \mathbb N:\forall n>l$

$$ p_{n}<n-nW_{-1}\bigg(\frac{-e^{2}}{n}\bigg)$$

C. AXLER in [1] showed that $ \exists \ \ l \in \mathbb N:\forall n>l$ :

$ p_{n}<n\Big(\ln(n)+\ln_{2}(n)-1+\frac{\ln_{2}(n)-2}{\ln(n)}-\frac{\ln_{2}(n)^{2}-6\ln_{2}(n)+10.667}{2\ln(n)^{2}}\Big)=\Theta_{n}$ $ p_{n}>n\Big(\ln(n)+\ln_{2}(n)-1+\frac{\ln_{2}(n)-2}{\ln(n)}-\frac{\ln_{2}(n)^{2}-6\ln_{2}(n)+10.667}{2\ln(n)^{2}}\Big)=\Phi_{n} $

So

$p_{n}=n\Big(\ln(p_{n}-n)-\ln(p_{n}-n)+\frac{p_{n}}{n}\Big)<n\Big(\ln(p_{n}-n)-\ln(\Phi_{n} -n)+\frac{\Theta_{n}}{n}\Big)$

$j_n=\ln(\Phi_{n} -n)-\frac{\Theta_{n}}{n}$

Since we have $j_n>1 \ $for $n\geq32$ (by computation)

$\ln(p_{n}-n)+\frac{p_{n}}{n}>j_n>1$

$p_{n}<n\Big(\ln(p_{n}-n)-j_n\Big)<n\Big(\ln(p_{n}-n)-1\Big)$.

$p_n+n<n\ln(p_{n}-n)$

$e^{\ \frac{p_n}{n}+1}<p_n-n$

$e^{2}e^{\ \frac{p_n}{n}-1}<n(\frac{p_n}{n}-1)$

$-\frac{e^{2}}{n}>-(\frac{p_n}{n}-1) e^{-(\ \frac{p_n}{n}-1)}$

$$ p_{n}<n-nW_{-1}\bigg(\frac{-e^{2}}{n}\bigg) $$

$ \ $

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[1] NEW ESTIMATES FOR THE n-TH PRIME NUMBER, CHRISTIAN AXLER, Jun 2017 https://arxiv.org/pdf/1706.03651.pdf

https://math.stackexchange.com/questions/3476635/prime-numbers-upper-bound-involving-lambert-w-proof

https://www.researchgate.net/publication/258373882_Primes_and_the_Lambert_W_function