Editing Sato–Tate conjecture (section)

==Refinements==
There are also more refined statements. The '''Lang–Trotter conjecture''' (1976) of [[Serge Lang]] and [[Hale Trotter]] states the asymptotic number of primes ''p'' with a given value of ''a''<sub>''p''</sub>,<ref>{{Citation |last1=Lang |first1=Serge |last2=Trotter |first2=Hale F. |year=1976 |title=Frobenius Distributions in GL<sub>2</sub> extensions |location=Berlin |publisher=Springer-Verlag |isbn=978-0-387-07550-1 }}</ref> the trace of Frobenius that appears in the formula. For the typical case (no [[complex multiplication]], trace ≠ 0) their formula states that the number of ''p'' up to ''X'' is asymptotically

:<math>c \sqrt{X}/ \log X\ </math>

with a specified constant ''c''. [[Neal Koblitz]] (1988) provided detailed conjectures for the case of a prime number ''q'' of points on ''E''<sub>''p''</sub>, motivated by [[elliptic curve cryptography]].<ref>{{Citation |last=Koblitz |first=Neal |year=1988 |title=Primality of the number of points on an elliptic curve over a finite field |journal=Pacific Journal of Mathematics |volume=131 |issue=1 |pages=157–165 |mr=0917870 |doi=10.2140/pjm.1988.131.157|doi-access=free }}.</ref>
In 1999, [[Chantal David]] and [[Francesco Pappalardi]] proved an averaged version of the Lang–Trotter conjecture.<ref name="david-pappalardi">{{cite news |url=https://cms.math.ca/MediaReleases/2013/kn-prize.html |archiveurl=https://web.archive.org/web/20170201044745/https://cms.math.ca/MediaReleases/2013/kn-prize.html |archivedate=2017-02-01 |date=2013-04-15 |title=Concordia Mathematician Recognized for Research Excellence |access-date=2018-01-15 |work=[[Canadian Mathematical Society]] }}</ref><ref name="David Pappalardi distributions of elliptic curves">{{cite journal | last1=David | first1=Chantal | last2=Pappalardi | first2=Francesco | title=Average Frobenius distributions of elliptic curves | journal=International Mathematics Research Notices | volume=1999 | issue=4 | date=1999-01-01 | pages=165–183| doi=10.1155/S1073792899000082 }}</ref>