where the same summation notation is used and denotes the unique integer such that but . Heuristically, the first of these two functions the number of primes less than and the second does the same, but weighting each prime in accordance with their logarithmic relationship to .
Many innocuous results in number owe their proof to a relatively analysis of the asymptotics of one or both of these functions. For example, the fact that for any , we have
is equivalent to the statement that .
A somewhat less innocuous result is that the prime number theorem (i.e., that ) is equivalent to the statement that , which in turn, is equivalent to the statement that .
|Date of creation||2013-03-22 13:50:15|
|Last modified on||2013-03-22 13:50:15|
|Last modified by||Mathprof (13753)|