Page "Prime number theorem" Paragraph 10

Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1797 or 1798 that π ( a ) is approximated by the function a /( A ln ( a ) + B ), where A and B are unspecified constants.

In the second edition of his book on number theory ( 1808 ) he then made a more precise conjecture, with A = 1 and B = − 1. 08366.

Carl Friedrich Gauss considered the same question: " Ins Jahr 1792 oder 1793 ", according to his own recollection nearly sixty years later in a letter to Encke ( 1849 ), he wrote in his logarithm table ( he was then 15 or 16 ) the short note " Primzahlen unter ".

But Gauss never published this conjecture.

In 1838 Johann Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li ( x ) ( under the slightly different form of a series, which he communicated to Gauss ).

Both Legendre's and Dirichlet's's formulas imply the same conjectured asymptotic equivalence of π ( x ) and x / ln ( x ) stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.