Prime Quadratic Effect

Let $\pi_{m,n}(x)$ denote the number of Primes $\leq x$ which are congruent to $n$ modulo $m$. Then one might expect that

$$\Delta(x)\equiv\pi_{4,3}(x)-\pi_{4,1}(x)\sim {\textstyle{1\over 2}}\pi(x^{1/2})>0$$

(Berndt 1994). Although this is true for small numbers, Hardy and Littlewood showed that $\Delta(x)$ changes sign infinitely often. (The first number for which it is false is 26861.) 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 $\lim_{x\to\infty}\Delta(x)=\infty$ (Berndt 1994).

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 $4n\pm 1$.'' 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.