The Distribution of Primes and the Riemann Zeta Function
We recall that the problem of the distribution of primes had been raised at least as far back as the Greek antiquity. The proof of our Theorem 3.9, that there are infinitely many primes, appears in Euclid (Book 9, Section 20), and Eratosthenes† devised a systematic method for obtaining all primes up to any given number x.
KeywordsZeta Function Critical Line Riemann Zeta Function Riemann Hypothesis Integral Converge
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- 5.M. B. Barban, Doklady Akademii Nauk UzSSr 8 (1961) 9–11.Google Scholar
- 11.L. Euler, Institutiones Calculi Differentialis, Pt. 2, Chapters 5 and 6. St. Petersburg: Acad. Imper. Scient. Petropolitanae, 1755; Opera Omnia (1), vol. 10.Google Scholar
- 15.D. Jackson, Fourier series and Orthogonal Polynomials, Carus Monograph, No. 6. Menasha, Wise: G. Banta, 1941.Google Scholar
- 23.B. Riemann, Collected Works of B. Riemann, edited by H. Weber, 2nd ed. (1892/1902). New York: Dover Publishing, 1953.Google Scholar
- 24.J. B. Rosser, J. M. Yohe, L. Schoenfeld, Information Processing, 1968. Proc. IFIP Congress Edinburgh, 1968, vol. 1, pp. 70–76. Amsterdam: North Holland, 1969.Google Scholar
- 27.A. Selberg, The general sieve method...in prime number theory, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1 (1950) 286–292.Google Scholar
- 30a.Wang Yuan, Several papers on the representation of large integers as a sum of a prime and an almost prime; in particular, Acta Math. Sinica 10 (1960) 168–181;Google Scholar
- 30b.Wang Yuan, Several papers on the representation of large integers as a sum of a prime and an almost prime; in particular, Sciencia Sinica 11 (1962) 1033–1054 (this is essentially an English translation of the previous paper, plus a most interesting Appendix).Google Scholar
- 31.D. V. Widder, Advanced Calculus, 2nd ed. Englewood Cliffs, N.J.: Prentice-Hall, 1961.Google Scholar