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Prime counting function

The prime counting function is the function giving the number of primes less than or equal to a given number (Shanks 1993, p. 15). For example, there are no primes , so . There is a single prime (2) , so . There are two primes (2 and 3) , so . And so on.The notation for the prime counting function is slightly unfortunate because it has nothing whatsoever to do with the constant . This notation was introduced by number theorist Edmund Landau in 1909 and has now become standard. In the words of Derbyshire (2004, p. 38), "I am sorry about this; it is not my fault. You'll just have to put up with it."Letting denote the th prime, is a right inverse of since(1)for all positive integers. Also,(2)iff is a prime number.The first few values of for , 2, ... are 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, ... (OEIS A000720). The Wolfram Language command giving the prime counting function for a number is PrimePi[x], which works up to a maximum value of .The notation..

Lehmer's phenomenon

The appearance of nontrivial zeros (i.e., those along the critical strip with ) of the Riemann zeta function very close together. An example is the pair of zeros given by and , illustrated above in the plot of . This corresponds to the region near Gram point (Lehmer 1956; Edwards 2001, p. 178).Let be the th nontrivial root of , and consider the local extrema of . Then the values of after which the absolute value of the local extremum between and decreases are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, ... (OEIS A114886).

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