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Various citation cleanup (identifiers mostly), replaced: | id={{MR|2721467}} → | mr=2721467 using AWB

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In [[number theory]], '''Chebyshev's bias''' is the phenomenon that most of the time, there are more primes of the form 4''k'' + 3 than of the form 4''k'' + 1, up to the same limit. This phenomenon was first observed by [[Pafnuty Chebyshev|Chebyshev]] in 1853.

==Description==
Let π(''x''; 4, 1) denote the number of primes of the form 4''k'' + 1 up to ''x''. Similarly, let π(''x''; 4, 3) denote the number of primes of the form 4''k'' + 3 up to ''x''. By the prime number theorem, extended to arithmetic progression,

:<math>\pi(x;4,1)\sim\pi(x;4,3)\sim \frac{1}{2}\frac{x}{\log x},</math>

i.e., half of the primes are of the form 4''k'' + 1, and half of the form 4''k'' + 3. A reasonable guess would be that π(''x''; 4, 1) > π(''x''; 4, 3) and π(''x''; 4, 1) < π(''x''; 4, 3) each also occur 50% of the time. This, however, is not supported by numerical evidence — in fact, π(''x''; 4, 3) > π(''x''; 4, 1) occurs much more frequently. Indeed this
inequality holds for all primes ''x'' < 26833 except 5, 17, 41 and 461, for which there is a tie.

In general, if 0 < ''a'', ''b'' < ''q'' are integers, (''a'', ''q'') = (''b'', ''q'') = 1, ''a'' is a quadratic residue, ''b'' is a quadratic nonresidue mod ''q'', then π(''x''; ''q'', ''b'') > π(''x''; ''q'', ''a'') occurs more often than not. This has been proved only by assuming strong forms of the [[Riemann hypothesis]]. The conjecture of Knapowski and [[Pál Turán|Turán]], however, that the density of the numbers ''x'' for which π(''x''; 4, 3) > π(''x''; 4, 1) holds, is 1, turned out to be false. They, however, do have a [[natural density|logarithmic density]], which is approximately 0.9959....<ref>(Rubinstein—Sarnak, 1994)</ref>

==See also==
* [[Shanks–Rényi race problem]]

==References==
{{reflist}}
* P.L. Chebyshev: Lettre de M. le Professeur Tchébychev à M. Fuss sur un nouveaux théorème relatif aux nombres premiers contenus dans les formes 4''n'' + 1 et 4''n'' + 3, ''Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg'', '''11''' (1853), 208.
* {{cite journal
|first1=Andrew
|last1=Granville
|author1-link=Andrew Granville
|first2=Greg
|last2=Martin
|title=Prime number races
|journal=[[American Mathematical Monthly|Amer. Math. Monthly]]
|volume=113
|year=2006
|pages=1–33
|jstor=27641834}}
* J. Kaczorowski: On the distribution of primes (mod 4), ''Analysis'', '''15''' (1995), 159–171.
* S. Knapowski, Turan: Comparative prime number theory,I, ''[[Acta Mathematica Hungarica|Acta Math. Acad. Sci. Hung.]]'', '''13''' (1962), 299–314.
* {{Cite journal
|first1=M.
|last1= Rubinstein
|first2= P.
|last2= Sarnak
|title= Chebyshev's bias
|journal=[[Experimental Mathematics (journal)|Experimental Mathematics]]
|volume=3
|year= 1994
|pages= 173–197
}}
* {{MathWorld|title=Chebyshev Bias|id=ChebyshevBias}}
* {{OEIS|A007350}} (where prime race 4n-1 versus 4n+1 changes leader)

[[Category:Analytic number theory]]
[[Category:Theorems in number theory]]
[[Category:Prime numbers]]

Kronecker coefficient - Revision history

en>Headbomb: Various citation cleanup (identifiers mostly), replaced: | id={{MR|2721467}} → | mr=2721467 using AWB