 It can be shown that is related to the von Mangoldt function , and hence to , via the relation

A delicate analysis of this equation and related properties of the zeta function, using the Mellin transform and Perron's formula, shows that for non-integer the equation

holds, where the sum is over all zeros (trivial and nontrivial) of the zeta function.This striking formula is one of the so-called explicit formulas of number theory, and is already suggestive of the result we wish to prove, since the term (claimed to be the correct asymptotic order of ) appears on the right-hand side, followed by (presumably) lower-order asymptotic terms.The next step in the proof involves a study of the zeros of the zeta function.