The Gram series is a better approximation than this result for small values of n, and Skewes identified an upper bound for the first "crossing point" associated with this result; that crossing was generalized by Littlewood.  
The Gram series is a better approximation than this result for small values of n, and Skewes identified an upper bound for the first "crossing point" associated with this result; that crossing was generalized by Littlewood.  This result is equivalent to two limits involving the Chebyshev functions over x, and modern calculations are given by Lagarias and Odlyzko.  Legendre's formulation makes use of two free parameters, one of which is one and the other approximately negative one and sometimes called Legendre's Constant, while the current form uses Guass's estimate of the logarithmic integral.  It was proved independently by Hadamard and de la Vallee Poussin through the nontrivial zeros of the Riemann zeta function.  For 10 points, name this result that gives an asymtotic approximation for the namesake counting function, often symbolized by pi.
The Gram series is a better approximation than this result for small values of n, and Skewes identified an upper bound for the first "crossing point" associated with this result; that crossing was generalized by Littlewood.  
{Prime Number Theorem} [accept {Prime Counting Function} before "this result"]