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Bibliography Bibliography M R denotes reference to Mathematical Rerie11.·s. I. Apostol, 1om M. ( 1970) Euler's <p-function and separable Gauss sums. Proc. A mer. Math. Soc., 24: 482-485; MR 41. # 1661. 2. Apostol, Tom M. ( 1974) Mathematical Analysis, 2nd ed. Reading, Mass.: Addison­ Wesley Publishing Co. 3. Ayoub, Raymond G. (1963) An Introduction to the Analytic Theory of Numbers. Mathematical Surveys, No. 10. Providence, R. I.: American Mathematical Society. 4. Bell, E. T. (1915) An arithmetical theory of certain numerical functions. University of Washinqton Pub!. in Math. and Phys. Sci., No. 1, Vol. 1: 1-44. 5. Borozdkin, K. G. (1956) K voprosu o postoyanni I. M. Vinogradova. Trudy treteqo z·sesoluznoqo matematic'eskoqo siezda, Vol. I, Moskva [Russian]. 6. Buhstab, A. A. ( 1965) New results in the investigation of the Goldbach-Euler problem and the problem of prime pairs. [Russian]. Dokl. Akad. Nauk SSSR, 162: 735-738; MR 31, #2226. [English translation: (1965) Soviet Math. Dokl. 6: 729-732.) 7. Chandrasekharan, Komaravolu (1968) Introduction to Analytic Number Theory. Die Grundlehren der Mathematischen Wissenschaften, Band 148. New York: Springer-Verlag. 8. Chandrasekharan, Komaravolu (1970) Arithmetical Functions. Die Grundlehren der Mathematischen Wissenschaften, Band 167. New York. Springer-Verlag. 9. Chebyshev, P. L. Sur Ia fonction qui determine Ia totalite des nombres premiers inferieurs a une limite donee. (a) (1851) Mem. Ac. Sc. St. Ntershour,q. 6: 141-157. (b) (1852) Jour. de Math (I) /7: 341-365. [Oeuvres, 1: 27-48.] I 0. Chen, Jing-run ( 1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tonqhao (Foreign Lang. Ed.). 17: 385-386; MR 34, #7483. 329 Bibliography II. Clarkson, James A. (1966) On the series of prime reciprocals. Proc. Amer. Math. Soc., I7: 541; M R 32, # 5573. 12. Davenport, Harold ( 1967) Multiplicative Number Theory. Lectures in Advanced Mathematics, No. I. Chicago: Markham Publishing Co. 13. Dickson, Leonard Eugene (1919) History of the Theory of Numbers. (3 volumes). Washington, D. C.: Carnegie Institution of Washington. Reprinted by Chelsea Publishing Co., New York, 1966. 14. Dickson, Leonard Eugene (1930) Studies in the Theory of Numbers. Chicago: The University of Chicago Press. 15. Dirichlet, P. G. Lejeune (1837) Beweis des Satzes, dass jede unbegrenzte arith­ metische Progression, deren erstes Glied und Differenz ganze Zahlen ohne geme­ inschlaftichen Factor sind, unendliche viele Primzahlen enthalt. Abhand. Ak. Wiss. Berlin: 45-81. [Werke, I: 315-342.] 16. Dirichlet, P. G. Lejeune (1840) Ueber eine Eigenschaft der quadratischen Formen. Bericht Ak. Wiss. Berlin: 49-52. [Werke, I: 497-502.] 17. Edwards, H. M. (1974) Riemann's Zeta Function. New York and London: Academic Press. 18. Ellison, W. J. (1971) Waring's problem. Amer. Math. Monthly, 78: 10-36. 19. Erdos, Paul (1949) On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A., 35: 374-384: MR 10, 595. 20. Euler, Leonhard (1737) Variae observationes circa series infinitas. Commentarii Academiae Scientiarum Jmperialis Petropolitanae, Y: 160-188. [Opera Omnia (1), /4; 216-244.] 21. Euler, Leonhard (1748) lntroductio in Analysin Jnfinitorum, Vol. I. Lausanne: Bousquet. [Opera Omnia (1), 8.] 22. Franklin, F. (1881) Sur lc developpement du produit infini (I - x)(l - x 2 ) (I - x 3 )(1 - x4 ). • .. Comptes Rendus Acad. Sci. (Paris), 92: 448-450. 23. Gauss, C. F. (1801) Disquisitiones Arithmeticae. Lipsiae. [English translation: Arthur A. Clarke (1966) New Haven: Yale University Press. 24. Gauss, C. F. (1849) Letter to Encke, dated 24 December. [Werke, Vol. II, 444-447.] 25. Gerstenhaber, Murray (1963) The 152nd proof of the law of quadratic reciprocity. Amer. Math. Monthly, 70: 397--398; MR 27, # 100. 26. Goldbach, C. (1742) Letter to Euler, dated 7 June. 27. Grosswald, Emil (1966) Topics from the Theory of Numbers. New York: The Macmillan Co. 28. Hadamard, J. ( 1896) Sur Ia distribution des zeros de Ia fonction ((s) et ses conse­ quences arithmetiques. Bull. Soc. Math. France, 24: 199-220. 29. Hagis, Peter, Jr. (1973) A lower bound for the set of odd perfect numbers. Math. Comp., 27: 951-953; MR 48, #3854. 30. Hardy, G. H. (1940) Ramanujan. Twelve Lectures on Subjects Suyyested by His Life and Work. Cambridge: The University Press. 31. Hardy, G. H. and Wright, E. M. (1960) An Introduction to the Theory of Numbers, 4th ed. Oxford: Clarendon Press. 32. Herner, Ove (1954) Notes on the Diophantine equation y 2 - k = x3. Ark. Mat., 3: 67-77; MR 15,776. 33. Ingham, A. E. (1932) The Distribution of Prime Numbers. Cambridge Tracts in Mathematics and Mathematical Physics, No. 30. Cambridge: The University Press. 330 Bibliography 34. Jacobi. C. G. J. (1829) Fundamenta Nova Theoriae Functionum Ellipticarum. [Gesammelte Werke, Band I, 49-239.] 35. Kolesnik, G. A. ( 1969) An improvement of the remainder term in the divisor prob­ lem. (Russian). Mat. Zametki, 6: 545-554; MR 41, # 1659. [English translation: (1969), Math. Notes, 6: 784--791.] 36. Kruyswijk. D. (1950) On some well-known properties of the partition function p(n) and Euler's infinite product. Nieuw Arch. Wisk., (2) 23: 97-107; MR II, 715. 37. Landau, E. (1909) Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig: Teubner. Reprinted by Chelsea, 1953. 38. Landau, E. (1927) Vorlesungen iiber Zahlentheorie (3 volumes). Leipzig: Hirzel. Reprinted by Chelsea, 1947. 39. Leech, J. (1957) Note on the distribution of prime numbers. J. London Math. Soc., 32: 56-58: MR 18.642. 40. Legendre, A.M. (1798) Essai sur Ia Theorie des Nombres. Paris: Duprat. 41. Lehmer, D. H. (1959) On the exact number of primes less than a given limit. JllinoisJ. Math., 3: 381-388; MR 21, #5613. 42. Lehmer, D. H. (1936) On a conjecture of Ramanujan. J. London Math. Soc., 11: 114--118. 43. Lehmer, D. N. (1914) List of prime numbers from I to 10,006,721. Washington, D.C.: Carnegie Institution of Washington, Pub I. No. 165. 44. LeVeque, W. J. (1956) Topics in Number Theory (2 volumes). Reading, Mass.: Addison-Wesley Publishing Co. 45. LeVeque, W. J. (1974) Reviews in Number Theory (6 volumes). Providence, RI: American Mathematical Society. 46. Levinson, N. ( 1969) A motivated account of an elementary proof of the prime number theorem. Amer. Math. Monthly, 76: 225-245; MR 39, #2712. 47. Levinson, Norman ( 1974) More than one third of zeros of Riemann's zeta-function are on a = 1/2. Advances Math., 13: 383--436. 48. van Lint, Jacobus Hendricus (1974) Combinatorial Theory Seminar (Eindhoven University of Technology), Lecture Notes in Mathematics 382. Springer-Verlag, Chapter 4. 49. Littlewood, J. E. (1914) Sur Ia distribution des nombres premiers. Comptes Rendus Acad. Sci. (Paris), 158: 1869-1872. 50. Mills, W. H. (1947) A prime-representing function. Bull. Amer. Math. Soc., 53: 604; MR 8, 567. 51. Nevanlinna." V. (1962) Ober den elementaren Beweis des Primzahlsatzes. Soc. Sci. Fenn. Comment. Phys.-Math,. 27 No.3, 8 pp.; MR 26, #2416. 52. Niven, I. and Zuckerman, H. S. (1972) An Introduction to the Theory of Numbers, 3rd ed. New York: John Wiley and Sons, Inc. 53. Prachar, Karl (1957) Primzahlverteilun,q. Die Grundlehren der Mathematischen Wissenschaften, Band 91. Berlin-Gottingen-Heidelberg: Springer-Verlag. 54. Rademacher, Hans (1937) On the partition functionp(n). Proc. London Math. Soc., 43: 241-254. 55. Rademacher, Hans (1964) Lectures on Elementary Number Theory. New York: Blaisdell Publishing Co. 56. Rademacher, Hans (1973) Topics in Analytic Number Theory. Die Grundlehren der Mathematischen Wissenschaften, Band 169. New York-Heidelberg-Berlin: Springer-Verlag. 331 Bibliography 57. Renyi, A. ( 1948) On the representation of an even number as the sum of a single prime and a single almost-prime number. (Russian). b·. Akad. Nauk SSSR Ser. Mat., 12: 57-78; M R 9, 413. [English translation: ( 1962) A mer. Math. Soc. Trans/. 19 (2): 299-321.] 58. Riemann. B. (1859) Ober die Anzahl der Primzahlen unter einer gegebener Grosse. Monatsber. Akad. Berlin. 671-680. 59. Robinson, R. M. ( 1958) A report on primes of the form k · 2" + I and on factors of Fermat numbers. Proc. Amer. Math. Soc .. 9:673-681; MR 20, #3097. 60. Rosser. J. Barkley. and Schoenfeld. Lowell ( 1962) Approximate formulas for some functions of prime number theory. 11/inois J. Math., 6: 69-94; M R 25, # 1139. 61. Schnirelmann. L. (1930) On additive properties of numbers. (Russian). /zr. Don­ skmm Politechn. lnst. ( Nmi'Otscherkask), 14 (2-3): 3-28. 62. Selberg, A tie ( 1949) An elementary proof of the prime number theorem. Ann. of Math., 50: 305-313; MR 10,595. 63. Shanks. Daniel ( 1951) A short proof of an identity of Euler. Proc. A mer. Math. Soc .. 2: 747-749; MR 13, 321. 64. Shapiro. Harold N. ( 1950) On the number of primes less than or equal x. Proc. Amer. Math. Soc., 1: 346-348; MR 12, 80. 65. Shapiro, Harold N. (1952) On primes in arithmetic progression II. Ann. of Math. 52:231-243: MR 12, 81. 66. Shen, Mok-Kong ( 1964) On checking the Goldbach conjecture. Nordisk Tidskr. 1nformations-Behandling, 4: 243-245; M R 30, # 3051. 67. Sierpinski. Waclaw ( 1964) Element an· Theorv of Nwnhers. Translated from Polish by A. Hulanicki. Monografie Matematyczne, Tom 42. Warsaw:. Panstwowe Wzdawnictwo Naukowe. 68. Tatuzawa, Tikao, and Iseki Kaneshiro ( 1951) On Selberg's elementary proof of the prime number theorem. Proc. Japan Acad., 27: 340 342; MR 13,725. 69. Titchmarsh, E. C. (1951) The Theory of the Riemann Zeta Function. Oxford: Clarendon Press. 70. Uspensky, J. V., and Heaslett, M.A. (1939) Elementary Number Theory. New York: McGraw-Hill Book Co. 71. Vallee Poussin. Ch.
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