abstract analytic number theory adrien-marie legendre anton felkel arxiv arithmetic progression asymptotic analysis asymptotic expansion atle selberg big o notation carl friedrich gauss chebyshev function complex analysis coprime degree of a polynomial digital object identifier divisor elementary proof eric bach finite field fourier transform helge von koch if and only if international standard serial number irreducible polynomial jacques hadamard jeffrey shallit jurij vega landau prime ideal theorem logarithmic integral function lowell schoenfeld mathworld mellin transform modular arithmetic natural logarithm number theory on-line encyclopedia of integer sequences on the number of primes less than a given magnitude pierre dusart planetmath prime-counting function prime gap prime number primitive recursive arithmetic probability randomness riemann hypothesis riemann zeta function terence tao theorem university of tennessee at martin von mangoldt function