The explicit formula was first published by Riemann in 1859 and then proved
by von Mangoldt in 1895.
Initially, by applying Cauchy's Residue Theorem we get the explicit formula in this form: where the sum is over Note the factor of -1 in the relation which results in the signs of the residues being reversed. The above sum
separates into two sums, over the trivial and nontrivial zeros respectively.
The former sum becomes
x), giving
where the are now the "The 'explicit formula' suggests that there are connections between the
numbers (A.E. Ingham, There is another explicit formula worth noting, which relates to the "naive" prime counting function more directly: Just as - relates to , the function log() relates to the function which is given by the sum . These variants - and
log() on the zeta function are
the generating functions
of and
Recall that The relevant explicit formula (also due to Riemann/von Mangoldt) is given by Here relates to just as relates to . It should also be noted that in both explicit formulas, the sum over
nontrivial zeros of zeta is not back to proof outline archive tutorial mystery new search home contact |