= Chebyshev estimate from central binomial coefficients
{c}
The central <binomial coefficient> satisfies $4^m/(2m+1)\leq\binom{2m}{m}\leq4^m$. Prime factors between $m$ and $2m$ give a dyadic upper bound for the <Chebyshev theta function>. The prime-power exponents in the factorial quotient are sums of zeros and ones, so the central coefficient gives a linear lower bound for the <Second Chebyshev function>. Higher prime powers contribute only $O(\sqrt x\log x)$; hence the prime-counting function has upper and lower bounds of order $x/\log x$.
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