Editing Prime number theorem (section)

=== Prime number race ===
[[File:Chebyshev bias.svg|thumb|Plot of the function <math>\ \pi(x;4,3)-\pi(x;4,1) \ </math> for {{math|''n'' ≤ {{val|30000}}}}]]

Although we have in particular
: <math>\pi_{4,1}(x) \sim \pi_{4,3}(x) \ ,</math>
empirically the primes congruent to 3 are more numerous and are nearly always ahead in this "prime number race"; the first reversal occurs at {{math|''x'' {{=}} 26861}}.<ref name="Granville Martin MAA">
{{cite journal
 | doi = 10.2307/27641834
 | last1 = Granville | first1 = Andrew | author1-link = Andrew Granville
 | last2 = Martin | first2 = Greg
 | year = 2006
 | title = Prime number races
 | journal = [[American Mathematical Monthly]]
 | volume = 113 | issue = 1 | pages = 1–33
 | jstor = 27641834 | mr = 2202918
 | url = http://www.dms.umontreal.ca/%7Eandrew/PDF/PrimeRace.pdf
}}</ref>{{Rp|1–2}} However Littlewood showed in 1914<ref name="Granville Martin MAA"/>{{Rp|2}} that there are infinitely many sign changes for the function
: <math>\pi_{4,1}(x) - \pi_{4,3}(x) ~,</math>
so the lead in the race switches back and forth infinitely many times. The phenomenon that {{math|''π''<sub>4,3</sub>(''x'')}} is ahead most of the time is called [[Chebyshev's bias]]. The prime number race generalizes to other moduli and is the subject of much research; [[Pál Turán]] asked whether it is always the case that {{math|''π''<sub>''c'',''a''</sub>(''x'')}} and {{math|''π''<sub>''c'',''b''</sub>(''x'')}} change places when {{mvar|a}} and {{mvar|b}} are coprime to {{mvar|c}}.<ref name=GuyA4>{{cite book |last=Guy | first=Richard K. | author-link=Richard K. Guy | year=2004 | title=Unsolved Problems in Number Theory | publisher=[[Springer-Verlag]] |edition=3rd |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=§A4, p. 13–15}}  This book uses the notation {{math|''π''(''x'';''a'',''c'')}} where this article uses {{math|''π''<sub>''c'',''a''</sub>(''x'')}} for the number of primes congruent to {{mvar|a}} modulo {{mvar|c}}.</ref> [[Andrew Granville|Granville]] and Martin give a thorough exposition and survey.<ref name="Granville Martin MAA" />
[[File:Prime race of last digit up to 10000.png|thumb|Graph of the number of primes ending in 1, 3, 7, and 9 up to {{math|''n''}} for {{math|''n'' < {{val|10000}}}}]]
Another example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits. However, empirical evidence shows that, for a given limit, there tend to be slightly more primes that end in 3 or 7 than end in 1 or 9 (a generation of the Chebyshev's bias).<ref>{{cite journal |last1=Lemke Oliver |first1=Robert J. |last2=Soundararajan |first2=Kannan |date=2016-08-02 |title=Unexpected biases in the distribution of consecutive primes |journal=Proceedings of the National Academy of Sciences |language=en |volume=113 |issue=31 |pages=E4446-54 |doi=10.1073/pnas.1605366113 |doi-access=free |issn=0027-8424 |pmc=4978288 |pmid=27418603 |arxiv=1603.03720 |bibcode=2016PNAS..113E4446L }}</ref> This follows that 1 and 9 are [[quadratic residue]]s modulo 10, and 3 and 7 are quadratic nonresidues modulo 10.