“Prime Quadratic Effect”的概念、定义、习题解题方法及翻译-数学辞典

单词

Prime Quadratic Effect

释义

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 .'' 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 .'' 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.

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