Nearest-neighbor-spacing distribution of prime numbers and quantum chaos

We give heuristic arguments and    computer results to support the hypothesis that, after appropriate rescaling, the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the oscillations in the nearest-neighbor-spacing distribution of primes. These oscillations have the very profound period of length six. We also calculate the spectral rigidity ${\Delta }_3$ for prime numbers by two methods. After suitable averaging one of these methods gives the Poisson dependence ${\Delta }_3\left(L\right)=L/15$.

Figure 1

Plots of ${\tau }_d\left(x\right)$ for $x=2^{24},2^{26},...,2^{46},2^{48}$. Wider lines represent exponential fits $a\left(x\right)e^{-db\left(x\right)}$ to ${\tau }_d\left(x\right)$. In the inset the plots of ${\tau }_d\left(x\right)/P\left(d\right)$ are shown.

Figure 2

The plot of power spectrum $S\left(f\right)$ calculated from $M=2^{10}=1024$ values of $P\left(d\right)$ plotted versus $1/f$ to show main periods of $P\left(d\right)$. The $y$ axis was broken to make visible peaks at $d\ne 6$. In the inset the plots of $P\left(d\right)$ and the approximation (14) are presented.

Figure 3

The comparison of $G\left(x\right)$ (black solid line) obtained from the computer search (up to $x=2^{48}$ we have used our own data, and for larger $x$ we took data from the Web pages [30]). For the plot of (15) [red (dark gray) dashed line] we have used the tabulated values of $\pi \left(x\right)$ available at Ref. [30]. The plot of the Cramer conjecture is also presented [blue (light gray) dash-dotted line].

Figure 4

The plot of histograms of unfolded spacings $D_n=r_{n+1}-r_n$ where $r_n=\mathrm{Li}\left(p_n\right)$ for primes up to $2^{34}=1.72...\times {10}^{10}$. Three widths of bins are used: $\Delta D=0.1$ [green (light gray)], $\Delta D=0.001$ [blue (medium gray)], and $\Delta D=16.54/28000$ [red (dark gray)]. In black (below the legend) is shown the plot for the unfolding defined by Eq. (19).

Figure 5

Plots of $(\mathcal{D}(x,d),{\mathcal{T}}_d\left(x\right)),(d=2,4,...)$ for $x=2^{28}$ (the bottom, shortest, plot), $2^{38},2^{48}$ and in red (the longest) the plot of $e^{-u}$ (the order of plots corresponds to the order of descriptions in the legend). Only the points with ${\tau }_d\left(x\right)>1000$ were plotted to avoid fluctuations at large $D(x,d)$ due to small values of ${\tau }_d\left(x\right)$ for large $d$.

Figure 6

Plot of slopes $s\left(x\right)$ and prefactors $a\left(x\right)$ in the dependence $a\left(x\right)e^{-s\left(x\right)}$ obtained from fitting it to $(\mathcal{D}(d,x),ln\left({\mathcal{T}}_d\left(x\right)\right))$ for $x=2^{28},2^{29},...,2^{48}$.

Figure 7

Plots of ${\Delta }_3(x;L)$ obtained from (21) for $x={10}^8$ (triangles), $x={10}^9$ (circles), and $x={10}^{10}$ (squares) and $L=2^7=128,...,2^{26}=67108864$.

Figure 8

The plot of ${\Delta }_3^{\left(p\right)}\left(L\right)$ for probabilistic primes for one particular realization of the “artificial primes” in green (boxes) and averaged over 100 samples in black (circles). The last plot perfectly coincides with the red (dark gray) line representing $L/15$. In blue (light gray) is the fit $0.061L$ to circles plotted (practically indistinguishable from straight line $L/15=0.066...\times L$).

Figure 9

Plots of ${\Delta }_3^{\prime }(x_1;L;h)$ for $x_1={10}^{13}$ and $L=16h_1=1.6\times {10}^{10},32h_1=3.2\times {10}^{10},...2^{23}{h}_1=8.388608\times {10}^{15}$ and three values of $h_1={10}^9$ (black triangles), $h_2=2h_1$ (blue circles) and $h_3=4h_1$ (red squares). On the right in purple (no symbol) are plotted values of the number of terms $L/h_i-1$ in the sum (27) and the right $y$ axis also in purple (no symbol) is for these numbers. The dashed lines represent values of (31). The coincidence of ${\Delta }_3^{\prime }(x_1;L)$'s for all $h_i$ starts at approximately $L=2^{14}{h}_1=1.6384\times {10}^{13}$.

Figure 10

Plots of ${\Delta }_3^{\prime }(x_2;L;h)$ for $x_1={10}^{16}$ and $L=16h_1=1.6\times {10}^{10},32h_1=3.2\times {10}^{10},...2^{23}{h}_1=8.388608\times {10}^{15}$ and additionally for $L=1.5\times {10}^{16}$ and three values of $h_1={10}^9$ (black triangles), $h_2=2h_1$ (blue circles), and $h_3=4h_1$ (red squares). On the right in purple (medium gray) are the plotted values of the number of terms $L/h_i-1$ in the sum (27) and the right $y$ axis, also in purple, is for these numbers. The dashed lines represent values of (31). The coincidence of ${\Delta }_3^{\prime }(x_2;L)$'s for all $h_i$ starts at approximately $L=2^{18}{h}_1=2.62144\times {10}^{14}$ and follows practically powerlike increase given by equation $2.5695\times {10}^{-37}{L}^{3.685}$. The green (light gray) line presents this equation multiplied by 100; however, we expect bending of ${\Delta }_3^{\prime }(x_2;L;h)$ for $L>x_2$ similar to the one seen in Fig. 9.

Figure 11

The illustration of the experimental fact that the straight line best fitting $(x+\epsilon )/ln(x+\epsilon )$ on the interval $\epsilon \in (0,L)$ crosses it at $\epsilon =L/4$ and $\epsilon =3L/4$.