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Monte-Carlo Approximation of the Prime Counting Function

This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum_\rho {\rm Li}\,(x^\rho)$.

Figure 3 shows that for $x=10^{12}$ the error $\delta\pi(x)$ is about 1100 for $N=10^3$ and 300 for $N=10^5$. The error drops by roughly a factor of two when the number of zeros is increased by a factor of ten. More extensive numerical data, with $x$ in the range $10^{10}$ to $10^{17}$, indicates that to reach the precise value of $\pi(x)$ one needs to include about $x^{1.38}$ zeros.

The value $x=10^6$ in the OP is well below this range, but as a first estimate I would conclude that to obtain $\pi(10^6)=78498$ to all five decimal places one would need $N\simeq 10^8$ zeros.

Monte-Carlo Approximation of the Prime Counting Function

This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum_\rho {\rm Li}\,(x^\rho)$.

Figure 3 shows that for $x=10^{12}$ the error $\delta\pi(x)$ is about 1100 for $N=10^3$ and 300 for $N=10^5$. The error drops by roughly a factor of two when the number of zeros is increased by a factor of ten. More extensive numerical data, with $x$ in the range $10^{10}$ to $10^{17}$, indicates that to reach the precise value of $\pi(x)$ one needs to include about $x^{1.38}$ zeros.

The value $x=10^6$ in the OP is well below this range, but as a first estimate I would conclude that to obtain $\pi(10^6)=78498$ accurate to the last of its five digits one would need $N\simeq 10^8$ zeros.

Monte-Carlo Approximation of the Prime Counting Function

This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum_\rho {\rm Li}\,(x^\rho)$.

Figure 3 shows that for $x=10^{12}$ the error $\delta\pi(x)$ is about 1100 for $N=10^3$ and 300 for $N=10^5$. The error drops by roughly a factor of two when the number of zeros is increased by a factor of ten. More extensive numerical data indicates that to reach a sub-unit accuracy in $\pi(x)$ one needs to include about $x^{1.38}$ zeros.

An empirical formula, fitted to data for $x$ between $10^{10}$ and $10^{17}$ and $N$ between 100 and $10^5$, is $$\log_2\delta\pi(x)=16.64-1.24\log_{10}N.$$

Monte-Carlo Approximation of the Prime Counting Function

This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum_\rho {\rm Li}\,(x^\rho)$.

Figure 3 shows that for $x=10^{12}$ the error $\delta\pi(x)$ is about 1100 for $N=10^3$ and 300 for $N=10^5$. The error drops by roughly a factor of two when the number of zeros is increased by a factor of ten. More extensive numerical data, with $x$ in the range $10^{10}$ to $10^{17}$, indicates that to reach the precise value of $\pi(x)$ one needs to include about $x^{1.38}$ zeros.

The value $x=10^6$ in the OP is well below this range, but as a first estimate I would conclude that to obtain $\pi(10^6)=78498$ to all five decimal places one would need $N\simeq 10^8$ zeros.

Monte-Carlo Approximation of the Prime Counting Function

This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum_\rho {\rm Li}\,(x^\rho)$.

Figure 3 shows that for $x=10^{12}$ the error $\delta\pi(x)$ is about 1100 for $N=10^3$ and 300 for $N=10^5$. The error drops by roughly a factor of two when the number of zeros is increased by a factor of ten. More extensive numerical data indicates that to reach a sub-unit accuracy in $\pi(x)$ one needs to include about $x^{1.38}$ zeros.

The heuristic formula, tested for $x$ between $10^{10}$ and $10^{17}$ and $N$ between 100 and $10^5$, is $$\log_2\delta\pi(x)=16.64-1.24\log_{10}N.$$

Monte-Carlo Approximation of the Prime Counting Function

This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum_\rho {\rm Li}\,(x^\rho)$.

Figure 3 shows that for $x=10^{12}$ the error $\delta\pi(x)$ is about 1100 for $N=10^3$ and 300 for $N=10^5$. The error drops by roughly a factor of two when the number of zeros is increased by a factor of ten. More extensive numerical data indicates that to reach a sub-unit accuracy in $\pi(x)$ one needs to include about $x^{1.38}$ zeros.

An empirical formula, fitted to data for $x$ between $10^{10}$ and $10^{17}$ and $N$ between 100 and $10^5$, is $$\log_2\delta\pi(x)=16.64-1.24\log_{10}N.$$