2
$\begingroup$

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]
$\endgroup$
1
  • 1
    $\begingroup$ So $li^{-1}(z) = f(W(z))$ with $f$ analytic which means $(u+1)e^u = \frac{f'(u)}{\log f(u)}$ $\endgroup$
    – reuns
    Aug 13, 2017 at 16:29

2 Answers 2

1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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)

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

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

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

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

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

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

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

$ \ $

$ \ $

[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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.