Landau,s Formula
المؤلف:
Landau, E
المصدر:
"Über die Nullstellen der Zetafunction." Math. Ann. 71
الجزء والصفحة:
...
9-9-2019
1534
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. http://www.ams.org/notices/200303/fea-conrey-web.pdf.
Landau, E. "Über die Nullstellen der Zetafunction." Math. Ann. 71, 548-564, 1911.
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