Chebyshev's bias

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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 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Chebyshev in 1853.

Description[edit]

Let π(xnm) denote the number of primes of the form nk + m up to x. By the prime number theorem, extended to arithmetic progression,

i.e., half of the primes are of the form 4k + 1, and half of the form 4k + 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 π(x; 4, 1) = π(x; 4, 3). Besides, the first prime x such that π(x; 4, 1) > π(x; 4, 3) is 26861, i.e. π(x; 4, 3) ≥ π(x; 4, 1) for all primes x < 26861.

In general, if 0 < a, b < n are integers, GCD(an) = GCD(bn) = 1, a is a quadratic residue mod n, b is a quadratic nonresidue mod n, then π(xnb) > π(xna) occurs more often than not. This has been proved only by assuming strong forms of the Riemann hypothesis. The conjecture of Knapowski and Turán, however, the stronger conjecture that the density of the numbers x for which π(x; 4, 3) > π(x; 4, 1) holds is 1 (i.e. π(x; 4, 3) > π(x; 4, 1) holds for almost all x), turned out to be false. They, however, do have a logarithmic density, which is approximately 0.9959....[1]

Generalizations[edit]

This is for k = −4 to find the smallest prime p such that (where is the kronecker symbol), however, for a given nonzero integer k (not only k = −4), we can also find the smallest prime p satisfying this condition. By the prime number theorem, for every nonzero integer k, there are infinitely many primes p satisfying this condition.

For positive integers k = 1, 2, 3, ..., the smallest primes p are

2, 11100143, 61981, 3, 2082927221, 5, 2, 11100143, 2, 3, 577, 61463, 2083, 11, 2, 3, 2, 11100121, 5, 2082927199, 1217, 3, 2, 5, 2, 17, 61981, 3, 719, 7, 2, 11100143, 2, 3, 23, 5, 11, 31, 2, 3, 2, 13, 17, 7, 2082927199, 3, 2, 61463, 2, 11100121, 7, 3, 17, 5, 2, 11, 2, 3, 31, 7, 5, 41, 2, 3, ... (OEISA306499 is a subsequence, for k = 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, ... OEISA003658)

For negative integers k = −1, −2, −3, ..., the smallest primes p are

2, 3, 608981813029, 26861, 7, 5, 2, 3, 2, 11, 5, 608981813017, 19, 3, 2, 26861, 2, 643, 11, 3, 11, 31, 2, 5, 2, 3, 608981813029, 48731, 5, 13, 2, 3, 2, 7, 11, 5, 199, 3, 2, 11, 2, 29, 53, 3, 109, 41, 2, 608981813017, 2, 3, 13, 17, 23, 5, 2, 3, 2, 1019, 5, 263, 11, 3, 2, 26861, ... (OEISA306500 is a subsequence, for k = −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31, −35, −39, −40, −43, −47, −51, −52, −55, −56, −59, ... OEISA003657)

For every (positive or negative) nonsquare integer k, there are more primes p with than with (up to the same limit) more often than not. If strong forms of the Riemann hypothesis are true.

Extension to higher power residue[edit]

Let m and n be integers such that m≥0, n>0, GCD(m, n) = 1, define a function , where is the Euler's totient function.

For example, f(1, 5) = f(4, 5) = 1/2, f(2, 5) = f(3, 5) = 0, f(1, 6) = 1/2, f(5, 6) = 0, f(1, 7) = 5/6, f(2, 7) = f(4, 7) = 1/2, f(3, 7) = f(5, 7) = 0, f(6, 7) = 1/3, f(1, 8) = 1/2, f(3, 8) = f(5, 8) = f(7, 8) = 0, f(1, 9) = 5/6, f(2, 9) = f(5, 9) = 0, f(4, 9) = f(7, 9) = 1/2, f(8, 9) = 1/3.

It is conjectured that if 0 < a, b < n are integers, GCD(an) = GCD(bn) = 1, f(a, n) > f(b, n), then π(xnb) > π(xna) occurs more often than not.

See also[edit]

References[edit]

  1. ^ (Rubinstein—Sarnak, 1994)
  • P.L. Chebyshev: Lettre de M. le Professeur Tchébychev à M. Fuss sur un nouveaux théorème relatif aux nombres premiers contenus dans les formes 4n + 1 et 4n + 3, Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg, 11 (1853), 208.
  • Granville, Andrew; Martin, Greg (2006). "Prime number races". Amer. Math. Monthly. 113: 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 Math. Acad. Sci. Hung., 13 (1962), 299–314.
  • Rubinstein, M.; Sarnak, P. (1994). "Chebyshev's bias". Experimental Mathematics. 3: 173–197. doi:10.1080/10586458.1994.10504289.

External links[edit]