Prime Quadratic Effect
Let pi_(m,n)(x) denote the number of primes <=x which are congruent to n modulo m (i.e., the modular prime counting function). Then one might expect that
| Delta(x)=pi_(4,3)(x)-pi_(4,1)(x)∼1/2pi(x^(1/2))>0 |
(Berndt 1994).
Although this is true for small numbers, Hardy and Littlewood showed that Delta(x) changes sign infinitely often. The effect was first noted by Chebyshev in 1853, and is sometimes called the Chebyshev phenomenon. It was subsequently studied by Shanks (1959), Hudson (1980), and Bays and Hudson (1977, 1978, 1979). The effect was also noted by Ramanujan, who incorrectly claimed that lim_(x->infty)Delta(x)=infty (Berndt 1994).
The bias of the sign of pi_(4,3)(n)-pi_(4,1)(n) is known as the Chebyshev bias.
See also
Chebyshev Bias, Modular Prime Counting FunctionExplore with Wolfram|Alpha
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References
Bays, C. and Hudson, R. H. "The Mean Behavior of Primes in Arithmetic Progressions." J. reine angew. Math. 296, 80-99, 1977.Bays, C. and Hudson, R. H. "On the Fluctuations of Littlewood for Primes of the Form 4n+/-1." Math. Comput. 32, 281-286, 1978.Bays, C. and Hudson, R. H. "Numerical and Graphical Description of All Axis Crossing Regions for the Moduli 4 and 8 which Occur Before 10^(12)." Internat. J. Math. Math. Sci. 2, 111-119, 1979.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 135-136, 1994.Hudson, R. H. "A Common Principle Underlies Riemann's Formula, the Chebyshev Phenomenon, and Other Subtle Effects in Comparative Prime Number Theory. I." J. reine angew. Math. 313, 133-150, 1980.Shanks, D. "Quadratic Residues and the Distribution of Primes." Math. Comput. 13, 272-284, 1959.Referenced on Wolfram|Alpha
Prime Quadratic EffectCite this as:
Weisstein, Eric W. "Prime Quadratic Effect." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrimeQuadraticEffect.html