Landau's Formula

Landau (1911) proved that for any fixed ,

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.

REFERENCES:

Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. https://www.ams.org/notices/200303/fea-conrey-web.pdf.

Landau, E. "Über die Nullstellen der Zetafunction." Math. Ann. 71, 548-564, 1911.