Editing Analytic number theory (section)

===Precursors===
Much of analytic number theory was inspired by the [[prime number theorem]]. Let π(''x'') be the [[prime-counting function]] that gives the number of primes less than or equal to ''x'', for any real number&nbsp;''x''. For example, π(10)&nbsp;=&nbsp;4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that ''x'' / ln(''x'') is a good approximation to π(''x''), in the sense that the [[limit of a function|limit]] of the ''quotient'' of the two functions π(''x'') and ''x'' / ln(''x'') as ''x'' approaches infinity is 1:

: <math>\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1,</math>

known as the asymptotic law of distribution of prime numbers.

[[Adrien-Marie Legendre]] conjectured in 1797 or 1798 that π(''a'') is approximated by the function ''a''/(''A'' ln(''a'')&nbsp;+&nbsp;''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''&nbsp;=&nbsp;1 and ''B''&nbsp;≈&nbsp;&minus;1.08366. [[Carl Friedrich Gauss]] considered the same question: "Im Jahr 1792 oder 1793" ('in the year 1792 or 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 <math>a(=\infty) \frac a{\ln a}</math>" ('prime numbers under <math>a(=\infty) \frac a{\ln a}</math>'). But Gauss never published this conjecture. In 1838 [[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 formulas imply the same conjectured asymptotic equivalence of π(''x'') and ''x''&nbsp;/&nbsp;ln(''x'') stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.