Prime Quadratic Effect
Let 
denote the number of Primes 
which are congruent to 
modulo 
. Then one mightexpect that
(Berndt 1994). Although this is true for small numbers, Hardy and Littlewood showed that
changes sign infinitelyoften. (The first number for which it is false is 26861.) The effect was first noted by Chebyshev in 1853, and issometimes called the Chebyshev Phenomenon. It was subsequently studied by Shanks (1959), Hudson (1980), and Baysand Hudson (1977, 1978, 1979). The effect was also noted by Ramanujan, who incorrectly claimed that
(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 Bays, C. and Hudson, R. H. ``Numerical and Graphical Description of All Axis Crossing Regions for the Moduli 4 and 8 which Occur Before 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.
.'' Math. Comput. 32, 281-286, 1978.
.'' Internat. J. Math. Math. Sci. 2, 111-119, 1979.
- Wiedersehen Manifold
- Wieferich Prime
- Wielandt's Theorem
- Wiener Filter
- Wiener Function
- Wiener-Khintchine Theorem
- Wiener Measure
- Wiener Space
- Wigner 3j-Symbol
- Wigner 6j-Symbol
- Wigner 9j-Symbol
- Wigner-Eckart Theorem
- Wilbraham-Gibbs Constant
- Wilcoxon Rank Sum Test
- Wilcoxon Signed Rank Test
- Wild Knot
- Wild Point
- Wilf-Zeilberger Pair
- Wilkie's Theorem
- Williams p+1 Factorization Method
- Wilson Plug
- Wilson Prime
- Wilson Quotient
- Douady's Rabbit Fractal
- Double Bubble