Five Compact Exponential Formulas for the Distribution of Primes

1. Notation

• $p_n$: the n-th prime.
• $\pi \left(x\right)$: the number of primes $\le x$.
• $p_n\#:={\prod }_{k=1}^n{p}_k$: the primorial of the first n primes.
• $ln\left(x\right)$: the natural logarithm.
• $W\left(z\right)$: Lambert’s W function, defined by $W\left(z\right)e^{W\left(z\right)}=z$.
• $\vartheta \left(n\right)={\sum }_{p\le n,p\mathrm{prime}}lnp$: the first Chebyshev function.

2. Five Key Formulas

2.1. (F1) ${\mathit{p}}_{\mathit{n}}$

$$p_n\approx {(nlnn)}^{1+{\displaystyle \frac{1}{{\pi }^{2}lnn}}-{\displaystyle \frac{1}{8{ln}^{2}n}}+{\displaystyle \frac{4/\pi }{{ln}^{3}n}}}$$

(1)

2.2. (F2) $\mathit{\pi }\left(\mathit{x}\right)$

$$\pi \left(x\right)\approx {\left({\displaystyle \frac{x}{lnx}}\right)}^{1-{\displaystyle \frac{1}{13lnx}}+{\displaystyle \frac{9/\pi }{{ln}^{2}x}}-{\displaystyle \frac{19/\pi }{{ln}^{3}x}}}$$

(2)

2.3. (F3) $ln({\mathit{p}}_{\mathit{n}}\#)$ from the prime index $\mathit{n}$

For indices $n\ge 100$, the logarithm of the primorial of the n-th prime is well approximated by:

$$ln(p_n\#)\approx {(ln(n!))}^{1+{\displaystyle \frac{1.301}{n}}}$$

(3)

2.4. (F4) Fast inverse $\mathit{p}\to \mathit{n}$

$$n\approx {\displaystyle \frac{p^{1/C\left(n_0\right)}}{ln\left(p^{1/C\left(n_0\right)}\right)}},\mathrm{where}C\left(t\right)=1+{\displaystyle \frac{1}{{\pi }^{2}lnt}}-{\displaystyle \frac{1}{8{ln}^{2}t}}+{\displaystyle \frac{4/\pi }{{ln}^{3}t}}\mathrm{and}{n}_0={\displaystyle \frac{p}{lnp}}.$$

(4)

A single evaluation of C gives $<0.5\%$ error for $p\ge 2500$.

2.5. (F5) Fast inverse $\mathit{\pi }\left(\mathit{x}\right)\to \mathit{x}$

$$x\approx {\displaystyle \frac{n^{1/E\left(x_0\right)}}{W\left(n^{1/E\left(x_0\right)}/E\left(x_0\right)\right)}},\mathrm{where}E\left(t\right)=1-{\displaystyle \frac{1}{13lnt}}+{\displaystyle \frac{9/\pi }{{ln}^{2}t}}-{\displaystyle \frac{19/\pi }{{ln}^{3}t}}\mathrm{and}{x}_0=nlnn.$$

(5)

Using the approximation $W\left(z\right)\simeq lnz-lnlnz$ keeps the relative error below $0.1\%$ for $n\ge 100$.

3. Links Between $ln(\mathit{n}!)$ and Prime-Sum Functions

A Stirling-based expansion relates the factorial to the Chebyshev function:

$$ln(n!)=\vartheta \left(n\right)+nlnlnn-n+{\displaystyle \frac{1}{2}}ln\left(2\pi n\right)+\mathcal{O}\left({\displaystyle \frac{nlnlnn}{lnn}}\right)$$

(6)

Alternatively, in an exponential style:

$$ln(n!)\approx {\left(\vartheta \left(n\right)\right)}^{1+{\displaystyle \frac{lnlnn-1}{lnn}}}$$

(7)

4. Numerical Accuracy (brief)

Empirical tests show:

• Formula (1): relative error $<0.1\%$ for $n\ge 100$.
• Formula (2): relative error $<0.7\%$ for ${10}^4\le x\le {10}^5$; $<0.2\%$ for $x\ge {10}^6$.
• Inverses (F4)–(F5): single-step accuracy $\lesssim 0.5\%$ (two steps $\lesssim 0.1\%$).

(Detailed tables and code are provided separately.)

5. Concluding Note on the Empirical Derivation

It is important to emphasize that while the formulas presented demonstrate remarkable accuracy in numerical tests, their deduction, including that of the constants, has been empirical in nature. This work is therefore presented as a motivation for the mathematical community. The theoretical foundation explaining this high effectiveness remains an open problem, as does establishing an analytical derivation for the coefficients, potentially connecting them to known mathematical constants.