The n^th Prime Exponentially

Abstract

Consider both the Logarithmic integral, $\mathrm{Li}\left(x\right)={lim}_{ϵ\to 0}\left\{{\int }_0^{1-ϵ}{\displaystyle \frac{du}{lnu}}+{\int }_{1+ϵ}^x{\displaystyle \frac{du}{lnu}}\right\}$, and the prime counting function $\pi \left(x\right)={\sum }_{p\le x}1$. From several recently developed known effective bounds on the prime counting function of the general form ${|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<ax(lnx)}^{b}exp\left(-c\sqrt{lnx}\right)$ for $x\ge x_0$ and known constants $\{a,b,c,x_0\}$, we shall show that it is possible to establish exponentially tight effective upper and lower bounds on the prime number theorem. For $x\ge x_{\ast }$, where $x_{\ast }\le max\{x_0,17\}$, we have the following: ${\displaystyle \frac{\mathrm{Li}\left(x\right)}{1+a{(lnx)}^{b+1}exp\left(-c\sqrt{lnx}\right)}}<\pi \left(x\right)<$ ${\displaystyle \frac{\mathrm{Li}\left(x\right)}{1-a{(lnx)}^{b+1}exp\left(-c\sqrt{lnx}\right)}}.$ These bounds provide a modern, and very clean and explicit, version of the celebrated prime number theorem. Furthermore, it is possible to establish exponentially tight effective upper and lower bounds on the location of the $n^{th}$ prime. Specifically, we find that $p_n<{\mathrm{Li}}^{-1}\left(n\left[1+a{(ln[nlnn])}^{b+1}exp\left(-c\sqrt{ln[nlnn]}\right)\right]\right)$ for $n\ge n_{\ast }$, whereas $p_n>{\mathrm{Li}}^{-1}\left(n\left[1-a{(ln[nlnn])}^{b+1}exp\left(-c\sqrt{ln[nlnn]}\right)\right]\right)$ for $n\ge n_{\ast }$. Herein, the range of validity is explicitly bounded by some calculable constant $n_{\ast }$ satisfying $n_{\ast }\le max\{\pi \left(x_0\right),\pi \left(17\right),\pi ((1+e^{-1})$ $exp\left({\left[{\displaystyle \frac{2(b+1)}{c}}\right]}^2\right)\left)\right\}.$ These bounds provide very clean and up-to-date and explicit information on the location of the $n^{th}$ prime number. Many other fully explicit bounds along these lines can easily be developed. Overall this article presents a general algorithmic approach to converting bounds on $\left|\pi \right(x)-\mathrm{Li}(x\left)\right|$ into somewhat clearer information regarding the primes.

1. Introduction: Known Bounds on $\left|\mathit{\pi }\right(\mathit{x})-\mathbf{Li}(\mathit{x}\left)\right|$

Work over the last decade or so has developed a number of fully effective and explicit bounds on the prime counting function $\pi \left(x\right)$ of the following general form [1,2,3]:

$${|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<ax(lnx)}^{b}exp\left(-c\sqrt{lnx}\right);(x\ge x_0),$$

(1)

where $\mathrm{Li}\left(x\right)$ is the well-known Logarithmic integral,

$$\mathrm{Li}\left(x\right)=\underset{ϵ\to 0}{lim}\left({\int }_0^{1-ϵ}{\displaystyle \frac{du}{lnu}}+{\int }_{1+ϵ}^x{\displaystyle \frac{du}{lnu}}\right)$$

(2)

and $\pi \left(x\right)$ is the prime counting function $\pi \left(x\right)={\sum }_{p\le x}1$. For some widely applicable effective bounds of this type, see

Table 1. Some widely applicable effective bounds on $\pi \left(x\right)$.

a	b	c	${\mathit{x}}_0$	Source	Notes
0.4394	−3/4	0.32115	59	Trudgian [1]	Equation (7)
0.2795	−3/4	0.3936	229	Trudgian [1]	Th 2
9.2211	1/2	0.8476	2	Fiori–Kadiri–Swidinsky [3]	Equations (3) and (43)
9.59	0.515	0.8274	2	Johnston–Yang [2]	Equation (1.6)

below. (A straightforward elementary numerical computation is required to determine the numerical coefficients in the Trudgian [1] bounds).

In the current article we shall develop algorithmic methods for converting such bounds on $\left|\pi \right(x)-\mathrm{Li}(x\left)\right|$ into practical and transparent information on the distribution and locations of the prime numbers. For instance the celebrated prime number theorem can be stated asymptotically as either $\pi \left(x\right)\sim \mathrm{Li}\left(x\right)$, or as $p_n\sim {\mathrm{Li}}^{-1}\left(n\right)$, and we will replace such asymptotic statements with explicit and effective upper and lower bounds.

2. Some Elementary Variants on These Bounds

There are various strategies for relaxing these bounds to make them more tractable, and more pragmatically useful. The most simple technique is to simply increase either of the coefficients $\{a,b\}$ or decrease the coefficient c in the Trudgian bounds so as to increase the domain of validity (decrease $x_0$). The FKS and JY bounds are already maximal in this regard. A more subtle technique (based on a variant of the discussion in reference [4]) is as follows: let $\tilde{b}\le b$ and $\tilde{c}<c$. Write

$$\begin{array}{lll}& & ax{(lnx)}^{b}exp\left(-c\sqrt{lnx}\right)\\ & & =ax\left\{{(lnx)}^{b-\tilde{b}}exp\left(-[c-\tilde{c}]\sqrt{lnx}\right)\right\}{(lnx)}^{\tilde{b}}exp\left(-\tilde{c}\sqrt{lnx}\right),\end{array}$$

(3)

then the quantity in braces is bounded, reaching a global maximum at

$$x_{max}=exp\left({\displaystyle \frac{4{[b-\tilde{b}]}^2}{{[c-\tilde{c}]}^2}}\right),$$

(4)

where it takes on the value

$$\begin{array}{ccc}\hfill {\left\{{(lnx)}^{b-\tilde{b}}exp\left(-[c-\tilde{c}]\sqrt{lnx}\right)\right\}}_{max}& =& \hfill {\left({\displaystyle \frac{4{[b-\tilde{b}]}^2}{{[c-\tilde{c}]}^2}}\right)}^{b-\tilde{b}}exp(-2[b-\tilde{b}]).\hfill \end{array}$$

(5)

Then, defining

$$\tilde{a}=a{\left({\displaystyle \frac{4{[b-\tilde{b}]}^2}{{[c-\tilde{c}]}^2}}\right)}^{b-\tilde{b}}exp(-2[b-\tilde{b}])=a{\left({\displaystyle \frac{2[b-\tilde{b}]}{e[c-\tilde{c}]}}\right)}^{2(b-\tilde{b})},$$

(6)

we see that

$$ax{(lnx)}^{b}exp\left(-c\sqrt{lnx}\right)\le \tilde{a}x{(lnx)}^{\tilde{b}}exp\left(-\tilde{c}\sqrt{lnx}\right);(x>1);$$

(7)

with equality only at $x=x_{max}$. Consequently, any bound of the form

$${|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<ax(lnx)}^{b}exp\left(-c\sqrt{lnx}\right);(x\ge x_0);$$

(8)

implies bounds of the form

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<\tilde{a}x{(lnx)}^{\tilde{b}}exp\left(-\tilde{c}\sqrt{lnx}\right);(x\ge x_0).$$

(9)

Here, the coefficients $\{\tilde{a},\tilde{b},\tilde{c}\}$ are as defined above. Both $\tilde{b}<b$ and $\tilde{c}<c$ are free to be chosen, while $\tilde{a}>a$ is to be derived. As long as $x_0$ is not too large, one can use explicit computations to find a larger domain of validity (some smaller ${\tilde{x}}_0$). Specifically, there will be some ${\tilde{x}}_0\le x_0$ such that

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<\tilde{a}x{(lnx)}^{\tilde{b}}exp\left(-\tilde{c}\sqrt{lnx}\right);(x\ge {\tilde{x}}_0;{\tilde{x}}_0\le x_0).$$

(10)

Note that because this procedure is intrinsically a relaxation (a weakening) of the bound, it is only mathematically worthwhile if this weakening is offset by either a significant increase in tractability (clarity) or an increase in the domain of validity.

In my previous work (see reference [4]), I focused on the case where $\tilde{b}=0$, which is the most appropriate for finding explicit bounds (for $\vartheta \left(x\right)$) of the de la Vallé Poussin form [5]. Herein, we shall soon see that for technical reasons (to be more fully explained below), when considering $\pi \left(x\right)$, rather than $\vartheta \left(x\right)$, it is more useful to consider the case $\tilde{b}=-1$.

Table 2. Several derived widely applicable effective bounds on $\pi \left(x\right)$. Input parameters on the left; output parameters on the right. Derived bounds are all of the form $|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<\tilde{a}x{(lnx)}^{\tilde{b}}exp\left(-\tilde{c}\sqrt{lnx}\right)$. Note the input and output bounds are closest to each other in the vicinity of $x_{max}$.

a	b	c	${\mathit{x}}_0$	$\tilde{\mathit{a}}$	$\tilde{\mathit{b}}$	$\tilde{\mathit{c}}$	${\mathit{x}}_{\mathit{max}}$	${\tilde{\mathit{x}}}_0$
0.4394	−3/4	0.32115	59	0.4680	−7/8	1/4	203,931	41
0.4394	−3/4	0.32115	59	0.4795	−1	1/6	35,439	41
0.2795	−3/4	0.3936	229	0.2804	−5/6	1/3	2097	227
0.2795	−3/4	0.3936	229	0.3164	−1	1/4	184,165	223
9.2211	1/2	0.8476	2	9.7590	0	1/2	3930	2
9.2211	1/2	0.8476	2	11.9026	−1/2	1/5	13,874	2
9.2211	1/2	0.8476	2	29.6698	−1	1/10	9,849,130	2
9.59	0.515	0.8274	2	11.148	0	1/2	19,877	2
9.59	0.515	0.8274	2	13.659	−1/2	1/5	35,206	2
9.59	0.515	0.8274	2	34.955	−1	1/10	34,331,213	2

below presents a number of derived widely applicable effective bounds of this general type. With Table 2 in hand, it is now possible to increase the parameter a to expand the range of validity. Several examples of this are given in

Table 3. More derived and very widely applicable effective bounds on $\pi \left(x\right)$, again of the form ${|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<ax(lnx)}^{b}exp\left(-c\sqrt{lnx}\right)$. Here, the parameter a is adjusted so as to maximize the region of validity, $x_0=2$.

a	b	c	${\mathit{x}}_0$
3/4	−3/4	0.32115	2
7/8	−7/8	0.32115	2
0.93	−1	0.32115	2
0.8935	−3/4	0.3936	2
0.94	−7/8	0.3936	2
1.05	−1	0.3936	2

.

We shall subsequently apply these bounds (both Table 2 and Table 3) in various ways—specifically to bounding the location of the $n^{th}$ prime number.

3. Exponentially Bounding the Prime Number Theorem

Note that for any $a>0$, $b\ge -1$, $c>0$, we can write

$$ax{(lnx)}^{b}exp\left(-c\sqrt{lnx}\right)=a{\displaystyle \frac{x}{lnx}}{(lnx)}^{b+1}exp\left(-c\sqrt{lnx}\right),$$

(11)

making it useful to define

$$f\left(x\right)={(lnx)}^{b+1}exp\left(-c\sqrt{lnx}\right).$$

(12)

Note the classic bound that ${\displaystyle \frac{x}{lnx}}<\pi \left(x\right)$ for $x>17$ [6], so that we can write

$$ax{(lnx)}^{b}exp\left(-c\sqrt{lnx}\right)<a\pi \left(x\right)f\left(x\right);(x>17).$$

(13)

This is already somewhat interesting since it implies that the various bounds on the quantity $\left|\pi \right(x)-\mathrm{Li}(x\left)\right|$ discussed above all lead to bounds of the form

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<a\pi \left(x\right)f\left(x\right);(x\ge max\{x_0,17\}).$$

(14)

When feasible (i.e., when $x_0$ is sufficiently small), a direct computation might potentially increase the domain of validity

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<a\pi \left(x\right)f\left(x\right);(x\ge x_{\ast };x_{\ast }\le max\{x_0,17\}).$$

(15)

This can be rearranged to yield

$${\displaystyle \frac{\mathrm{Li}\left(x\right)}{1+af\left(x\right)}}<\pi \left(x\right)<{\displaystyle \frac{\mathrm{Li}\left(x\right)}{1-af\left(x\right)}};(x\ge x_{\ast }),$$

(16)

where we note that $f\left(x\right)\to 0$ with exponential rapidity in $\sqrt{lnx}$. To be more explicit, for $x\ge x_{\ast }$, we have

$${\displaystyle \frac{\mathrm{Li}\left(x\right)}{1+a{(lnx)}^{b+1}exp\left(-c\sqrt{lnx}\right)}}<\pi \left(x\right)<{\displaystyle \frac{\mathrm{Li}\left(x\right)}{1-a{(lnx)}^{b+1}exp\left(-c\sqrt{lnx}\right)}},$$

(17)

with these two inequalities providing a remarkably clean and effective version of the prime number theorem.

4. Localizing the ${\mathit{n}}^{\mathit{th}}$ Prime Number

Note that for $b\ge -1$, the function $f\left(x\right)={(lnx)}^{b+1}exp(-c\sqrt{lnx})$ achieves a global maximum at $x_{peak}=exp\left({\left[2(b+1)/c\right]}^2\right)\ge 1$, above which $f^{\prime }\left(x\right)<0$, and below which $f^{\prime }\left(x\right)>0$.

Concentrating on the region above the peak, we shall first re-write $f\left(x\right)$ in terms of $p_{\pi \left(x\right)}$, yielding a bound in terms of $f\left(p_{\pi \left(x\right)}\right)$. We will then convert this into an explicit function of $\pi \left(x\right)$, yielding a bound in terms of $f\left(\pi \right(x)ln\pi (x\left)\right)$. Doing so will require giving some careful attention to the domain of validity of various inequalities. The aim is for the various manipulations introduced below to bring one closer to the peak without overshooting the peak.

First, observe that in all generality, $p_{\pi \left(x\right)}\le x<p_{\pi \left(x\right)+1}$; more specifically, one has $p_{\pi \left(x_{peak}\right)}\le x_{peak}<p_{\pi \left(x_{peak}\right)+1}$. Thus, if we now choose $x>p_{\pi \left(x_{peak}\right)+1}>x_{peak}$, then we can safely write

$$f\left(x\right)<f\left(p_{\pi \left(x\right)}\right);(x>p_{\pi (x_{peak})+1}).$$

(18)

Second, recall that in 1952, Nagura [7] (among other results) showed

$$\vartheta \left({\displaystyle \frac{4}{3}}x\right)-\vartheta \left(x\right)>0;(x\ge 109=p_{29}).$$

(19)

This bound is by no means optimal, but it will be more than good enough for current purposes. This implies

$$p_{n+1}<{\displaystyle \frac{4}{3}}{p}_n;(n\ge 29),$$

(20)

and explicitly checking smaller integers yields

$$p_{n+1}<{\displaystyle \frac{4}{3}}{p}_n;(n\ge 5=\pi \left(11\right)).$$

(21)

Let us now choose $x>{\displaystyle \frac{4}{3}}{x}_{peak}$ and note

$$x>{\displaystyle \frac{4}{3}}{x}_{peak}\ge {\displaystyle \frac{4}{3}}{p}_{\pi \left(x_{peak}\right)}>p_{\pi \left(x_{peak}\right)+1},$$

(22)

Then, we certainly have

$$f\left(x\right)<f\left(p_{\pi \left(x\right)}\right);\left(x>max\left\{11,{\displaystyle \frac{4}{3}}{x}_{peak}\right\}\right).$$

(23)

Now, invoke Rosser’s theorem $p_n>nlnn$, ($n\ge 1$) [8]; then,

$$f\left(x\right)<f\left(\pi \right(x)ln\pi (x\left)\right);$$

(24)

where, in addition to the prior constraints $x>max\{11,{\displaystyle \frac{4}{3}}{x}_{peak}\}$, we now also need to check that $\pi \left(x\right)ln\left(\pi \left(x\right)\right)>x_{peak}$.

To obtain a more computationally useful grasp on the domain of validity, use the fact that $p_n<nln(nlnn)$ for $n\ge 6$ [9] and note

$$p_n<nln(nlnn)=n(lnn+lnlnn)=nlnn\left\{1+{\displaystyle \frac{lnlnn}{lnn}}\right\};(n\ge 6),$$

(25)

where the quantity in braces is bounded and takes on the maximum value $1+e^{-1}$ at $x=e^e$. Thence,

$$p_n<(1+e^{-1})nlnn<1.367879442nlnn;(n\ge 4).$$

(26)

Thus, we see

$$\pi \left(x\right)ln\left(\pi \left(x\right)\right)>{\displaystyle \frac{p_{\pi \left(x\right)}}{1+e^{-1}}};(x\ge p_4=7)$$

(27)

This implies that the condition $\pi \left(x\right)ln\left(\pi \left(x\right)\right)>x_{peak}$ relevant to inequality (24) is certainly satisfied for $p_{\pi \left(x\right)}>(1+e^{-1})x_{peak}$, which in turn is certainly satisfied for the region $x>(1+e^{-1})x_{peak}>{\displaystyle \frac{4}{3}}{x}_{peak}$. Thus, we can safely write

$$f\left(x\right)<f(\pi \left(x\right)ln\pi \left(x\right));(x>max\{11,(1+e^{-1})x_{peak}\}).$$

(28)

Finally, combine this result with the conditions for the validity of the bound (14) to yield

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<a\pi \left(x\right)f(\pi \left(x\right)ln\pi \left(x\right));(x\ge max\{x_0,17,(1+e^{-1})x_{peak}\}),$$

(29)

where we remind the reader that for current purposes, we are interested in

$$x_{peak}=exp\left({\left[{\displaystyle \frac{2(b+1)}{c}}\right]}^2\right)\ge 1.$$

(30)

Because we have made various approximations to get to this stage, the actual range of the validity of the final inequality (29) may be somewhat larger than naively advertised—$max\{x_0,17,(1+e^{-1})x_{peak}\}$. That is, while the bounds (29) are certainly guaranteed to hold for values of x sufficiently far above the peak, they might still hold both at the peak and for some region below the peak.

Whenever feasible (meaning whenever $x_{peak}$ is not too large), this should be checked via an explicit computation. Evaluate $max\{x_0,17,(1+e^{-1})x_{peak}\}$ and explicitly check the inequality at all values of x below this location to find a suitable $x_{\ast }$ such that:

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<a\pi \left(x\right)f(\pi \left(x\right)ln\pi \left(x\right));(x\ge x_{\ast };x_{\ast }\le max\{x_0,17,(1+e^{-1})x_{peak}\}),$$

(31)

where sometimes $x_{\ast }$ is significantly lower than the naive bound $max\{x_0,17,(1+e^{-1})x_{peak}\}$. At other times, $(1+e^{-1})x_{peak}\le 17$, and in this situation the peak can be safely ignored. However, sometimes, we simply need to keep track of this complication.

If we evaluate this final inequality (31) at $x=p_n$, then

$$|\mathrm{Li}\left(p_n\right)-n|<anf(nlnn);(n\ge n_{\ast };n_{\ast }\le max\{\pi \left(x_0\right),7,\pi \left((1+e^{-1})x_{peak}\right)\}).$$

(32)

Finally, as promised in the abstract,

$${\mathrm{Li}}^{-1}\left(n[1-af(nlnn)]\right)<p_n<{\mathrm{Li}}^{-1}\left(n[1+af(nlnn)]\right);(n\ge n_{\ast }).$$

(33)

To be more explicit,

$$p_n<{\mathrm{Li}}^{-1}\left(n\left[1+a{(ln[nlnn])}^{b+1}exp\left(-c\sqrt{ln[nlnn]}\right)\right]\right);(n\ge n_{\ast });$$

$$p_n>{\mathrm{Li}}^{-1}\left(n\left[1-a{(ln[nlnn])}^{b+1}exp\left(-c\sqrt{ln[nlnn]}\right)\right]\right);(n\ge n_{\ast }).$$

Here, the range of validity is explicitly bounded by some $n_{\ast }$ with

$$n_{\ast }\le max\left\{\pi \left(x_0\right),7,\pi \left((1+e^{-1})exp\left({\left[{\displaystyle \frac{2(b+1)}{c}}\right]}^2\right)\right)\right\}.$$

These effective bounds can be viewed as fully explicit versions of the usual Cippola asymptotic expansion (see for instance the discussion in reference [10]).

See

Table 4. Some widely applicable effective bounds on $\left|\mathrm{Li}\right(x)-\pi (x\left)\right|$ and $p_n$ are as follows: ${|\mathrm{Li}\left(x\right)-\pi \left(x\right)|<a\pi \left(x\right)(ln[\pi \left(x\right)ln\pi \left(x\right)])}^{b+1}exp\left(-c\sqrt{ln\left[\pi \right(x)ln\pi (x\left)\right]}\right)$; and furthermore $p_n\lessgtr {\mathrm{Li}}^{-1}\left(n\left[1\pm a{(ln[nlnn])}^{b+1}exp\left(-c\sqrt{ln[nlnn]}\right)\right]\right)$.

a	b	c	${\mathit{x}}_0$	${\mathit{x}}_{\mathit{peak}}$	${\mathit{x}}_{\ast }$	${\mathit{n}}_0$	${\mathit{n}}_{\mathit{peak}}$	${\mathit{n}}_{\ast }$
0.4394	−3/4	0.32115	59	11.3	41	17	5	13
0.4680	−7/8	1/4	41	e	37	13	1	12
0.4795	−1	1/6	41	1	37	13	0	12
0.2795	−3/4	0.3936	229	5.022	149	50	3	35
0.2804	−5/6	1/3	227	e	149	49	1	35
0.3164	−1	1/4	223	1	97	48	0	25
9.2211	1/2	0.8476	2	275,789	2	1	24,104	1
9.7590	0	1/2	2	$e^{16}$	2	1	595,341	1
11.9026	−1/2	1/5	2	$e^{25}$	*	1	${\displaystyle \frac{1}{25}}{e}^{25}$	*
29.6698	−1	1/10	2	1	2	1	0	1
9.59	0.515	0.8274	2	667,161	2	1	54,105	1
11.148	0	1/2	2	$e^{16}$	2	1	595,431	1
13.659	−1/2	1/5	2	$e^{25}$	*	1	${\displaystyle \frac{1}{25}}{e}^{25}$	*
34.955	−1	1/10	2	1	2	1	0	1

Explicit evaluation of $x_{\ast }$ and $n_{\ast }$ in the situations labelled * proved computationally infeasible due to the magnitude of $x_{peak}$ and $n_{peak}$. All indications are that $x_{\ast }=2$ and $n_{\ast }=1$ in both cases.

below (based on Table 1, Table 2 and Table 3 above) for some explicit examples. The outputs from Table 2, $\{\tilde{a},\tilde{b},\tilde{c},{\tilde{x}}_0\}$, were relabelled as $\{a,b,c,x_0\}$ and used as inputs for new computations of $x_{\ast }$ and $n_{\ast }$ for the new inequalities discussed in this section.

5. Special Case

As previously indicated, the special case $b=-1$ is particularly appealing. Starting from any bound of the form

$${|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<ax(lnx)}^{-1}exp\left(-c\sqrt{lnx}\right);(x\ge x_0);$$

(34)

we have the particularly simple result that

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<a\pi \left(x\right)exp\left(-c\sqrt{lnx}\right);(x\ge x_{\ast };x_{\ast }\le max\{x_0,17\}),$$

(35)

whence in the range $x\ge x_{\ast }$, we have

$${\displaystyle \frac{\mathrm{Li}\left(x\right)}{1+aexp\left(-c\sqrt{lnx}\right)}}<\pi \left(x\right)<{\displaystyle \frac{\mathrm{Li}\left(x\right)}{1-aexp\left(-c\sqrt{lnx}\right)}},$$

(36)

and furthermore, noting that in this case we always have $x_{peak}=1$,

$$p_n<{\mathrm{Li}}^{-1}\left(n\left[1+aexp\left(-c\sqrt{ln[nlnn]}\right)\right]\right);(n\ge n_{\ast });$$

(37)

$$p_n>{\mathrm{Li}}^{-1}\left(n\left[1-aexp\left(-c\sqrt{ln[nlnn]}\right)\right]\right);(n\ge n_{\ast }),$$

(38)

which is subject to the simple constraint

$$n_{\ast }\le max\{\pi \left(x_0\right),7\}.$$

(39)

For several explicit effective examples of this specific behaviour, see

Table 5. Widely applicable effective bounds on $\left|\mathrm{Li}\right(x)-\pi (x\left)\right|$ and $p_n$ for $b=-1$ are as follows: $|\mathrm{Li}\left(x\right)-\pi \left(x\right)|<a\pi \left(x\right)exp\left(-c\sqrt{ln\left[\pi \right(x)ln\pi (x\left)\right]}\right)$; $p_n\lessgtr {\mathrm{Li}}^{-1}\left(n\left[1\pm aexp\left(-c\sqrt{ln[nlnn]}\right)\right]\right)$.

a	b	c	${\mathit{x}}_0$	${\mathit{x}}_{\mathit{peak}}$	${\mathit{x}}_{\ast }$	${\mathit{n}}_0$	${\mathit{n}}_{\mathit{peak}}$	${\mathit{n}}_{\ast }$
0.4795	−1	1/6	41	1	37	13	0	12
0.3164	−1	1/4	223	1	97	48	0	25
29.6698	−1	1/10	2	1	2	1	0	1
34.955	−1	1/10	2	1	2	1	0	1
1	−1	0.32115	2	1	2	1	0	1
1.1	−1	0.3936	2	1	2	1	0	1

(which is a subset of Table 4 and a minor extension of parts of Table 3). These bounds are not necessarily optimal but, given the initial input information in Table 1, are relatively simple to derive.

6. Further Developments

Another useful trick is to evaluate the inequality

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|<a\pi \left(x\right)f(\pi \left(x\right)ln\pi \left(x\right));(x\ge x_{\ast })$$

(40)

both at $x=p_n$ and at $x=p_{n+1}^{-}$, just below $p_{n+1}$. Then, since $\pi \left(p_{n+1}^{-}\right)=n=\pi \left(p_n\right)$, we have both

$$|n-\mathrm{Li}\left(p_n\right)|<anf(nlnn);(n\ge n_{\ast });$$

(41)

and

$$|n-\mathrm{Li}\left(p_{n+1}\right)|<anf(nlnn);(n\ge n_{\ast }).$$

(42)

However, by the triangle inequality,

$$\mathrm{Li}\left(p_{n+1}\right)-\mathrm{Li}\left(p_n\right)<2anf(nlnn);(n\ge n_{\ast });$$

(43)

which implies that

$$p_{n+1}<{\mathrm{Li}}^{-1}\left(\mathrm{Li}\left(p_n\right)+2anf(nlnn)\right);(n\ge n_{\ast }).$$

(44)

This observation can be used to develop more asymptotic bounds along the lines of the Cippola expansion, as discussed in [10]. It can also be related to the effective bounds developed using the first Chebyshev $\vartheta \left(x\right)$ function, as described in reference [11].

7. For the Future

One particular issue left for future investigation is the consideration of qualitatively different bounds of the form [2,3]

$$|\pi \left(x\right)-\mathrm{Li}\left(x\right)|\le 0.028x{(lnx)}^{0.801}exp\left(-0.1853{\displaystyle \frac{{(lnx)}^{3/5}}{{(lnlnx)}^{1/5}}}\right);(x\ge 23).$$

(45)

While such bounds asymptotically improve on any of the bounds we currently have under consideration, their algebraic structure is considerably more complex, sufficiently so as to make the discussion of this present article not directly applicable.

8. Conclusions

In hindsight, an ineffective version of the bounds considered herein could have been developed some 125 years ago. The current analysis is interesting in two respects: First, using developments made over the recent decade, the bounds were made explicitly and fully effective. Secondly, because the bounds are asymptotically exponential, they will eventually always overtake and outperform bounds based on the Cippola expansion. Finally, I do not claim any of these bounds are in any way optimal. As always, there is a three-way trade-off between stringency of the bound, simplicity and pragmatic usefulness of the bound, and the region of validity. Herein, I have largely focused on simplicity of the bound and a wide (sometimes maximal) region of validity.