Landau (1911) proved that for any fixed 
,
![sum_(0<|I[rho]|<=T)x^rho=-T/(2pi)Lambda(x)+O(lnT)](/images/equations/LandausFormula/NumberedEquation1.svg) |
as 
,
where the sum runs over the nontrivial Riemann
zeta function zeros and 
is the Mangoldt function.
Here, "fixed 
"
means that the constant implicit in 
depends on 
and, in particular, as 
approaches a prime or a prime power, the constant becomes
large.
Landau's formula is roughly the derivative of the explicit formula.
Landau's formula is quite extraordinary. If 
is not a prime or a prime
power, then 
and the sum grows as a constant times 
. But if 
is a prime or a prime
power, then 
and the sum grows much faster, like a constant times 
. This exhibits an amazing connection between the primes and
the 
s;
somehow the zeros "recognize" when 
is a prime and cause large contributions to the sum.
