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Adventures in arithmetick, or: How to make good use of a fourier transform

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Bibliography

  • Adleman, L. M., and Heath-Brown, D. R. (1985). The first case of Fermat’s Last Theorem,Invent. Math. 79, 409–416.

    Article  MATH  MathSciNet  Google Scholar 

  • Bateman, P. T. (1949). Note on the coefficients of the cyclotomic polynomial,Bull. Amer. Math. Soc. 55, 1180–1181.

    Article  MATH  MathSciNet  Google Scholar 

  • Brent, R. P., June, J. van de, Riele, H. J. T. te, and Winter, D.T. (1982). The first 200,000,001 zeros of Riemann’s zeta function.Computational methods in number theory, Part II, 384–403. Math. Centre Tracts, 155, Math Centrum, Amsterdam.

    Google Scholar 

  • Brun, V. (1920). Le crible d’Eratosthène et le théorème de Goldbach,Skrifter utgit av Videnskapsselskapet i Kristiania, mat.-naturv. Kl. 1:3.

  • Chen Jing Run and Pan Cheng Dong (1980). On the exceptional set of Goldbach numbers,Sci. Sinica 23, 219–232.

    Google Scholar 

  • Chen Jing Run (1983). On the exceptional set of Goldbach numbers (II),Sci. Sinica A, 327–342.

    Google Scholar 

  • Chudakov, N. G. (1938). On the density of the set of even integers which are not representable as a sum of two odd primes,Izv. Akad. Nauk SSSR, Ser. Mat. 1, 25–40.

    Google Scholar 

  • Corput, J. G. van der (1937). Sur l’hypothèse de Goldbach pour presque tous les nombres pairs,Acta Arith. 2, 266–290.

    Google Scholar 

  • Erdös, P. (1949). On the coefficients of the cyclotomic polynomial,Portugal. Math. 8, 63–71.

    MATH  Google Scholar 

  • Erdös, P. (1957). On the growth of the cyclotomic polynomial in the interval (0,1),Proc. Glasgow Math. Assoc. 3, 102–104.

    Article  MATH  MathSciNet  Google Scholar 

  • Estermann, T. (1938). Proof that almost all even positive integers are sums of two primes,Proc. London Math. Soc. 44, 307–314.

    Article  MathSciNet  Google Scholar 

  • Euler, L. (1742). Letter to Goldbach, 30 June 1742.Corresp. Math. Phys. (P. H. Fuss, ed.) 1 (1843), 135.

    Google Scholar 

  • Euler, L. (1849). De numeris amicabilibus,Comm. Arith. 2, 630. See alsoOpera postuma 1 (1862), 88.

    Google Scholar 

  • Fouvry, E. (1985). Théorème de Brun-Titchmarsh. Application au Théorème de Fermat,Invent. Math. 79, 383–407.

    Article  MATH  MathSciNet  Google Scholar 

  • Gauss, C. F. (1808). Summatio quarumdam serierum singularium,Werke II, 11–45.

    Google Scholar 

  • Goldbach, C. (1742). Letter to Euler, 7 June 1742.Corresp. Math. Phys. (P. H. Fuss, ed.) l (1843), 127.

    Google Scholar 

  • Goldstine, H. H., and von Neumann, J. (1953). A numerical study of a conjecture of Kummer.Math. Comp. 7, 133–143.

    Article  MATH  Google Scholar 

  • Hadamard, J. (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques,Bull. Soc. Math, de France 24, 199–220.

    MATH  MathSciNet  Google Scholar 

  • Hardy, G. H., and Littlewood, J. E. (1922). Some problems of ‘Partitio numerorum’ (III): On the expression of a number as a sum of primes,Acta Math. 44, 1–70.

    Article  MathSciNet  Google Scholar 

  • Hardy, G. H., and Littlewood, J. E. (1923). Some problems of ‘Partitio numerorum’ (V): A further contribution to the study of Goldbach’s problem,Proc. London Math. Soc. (2) 22, 46–56.

    MATH  MathSciNet  Google Scholar 

  • Heath-Brown, D. R. (1982). Prime numbers in short intervals and a generalized Vaughan identity,Can. J. Math. 34, 1365–1377.

    Article  MATH  MathSciNet  Google Scholar 

  • Heath-Brown, D. R., and Patterson, S. J. (1979). The distribution of Kummer sums at prime arguments,J. reine ang. Math. 310, 111–130.

    MATH  MathSciNet  Google Scholar 

  • Ingham, A. E. (1932).The distribution of prime numbers, Cambridge University Press.

  • Knuth, D. E. (1981).The art of computer programming. Vol. 2: Semi-numerical algorithms. Second edition. Addison-Wesley, Reading, Mass.

    Google Scholar 

  • Kummer, E. E. (1846). De residuis cubicis disquisitiones nonnullae analyticae,J. reine angew. Math. 32, 341–359 (=Coll. Papers, Vol. 1, 145-163, Springer-Verlag, Berlin, 1975).

    Article  MATH  Google Scholar 

  • Lehmer, E. (1936). On the magnitude of the coefficients of the cyclotomic polynomial,Bull. Amer. Math. Soc. 42, 389–392.

    Article  MathSciNet  Google Scholar 

  • Lehmer, E. (1956). On the location of Gauss sums,Math. Comp. 10, 194–202.

    Article  MATH  MathSciNet  Google Scholar 

  • Mangoldt, H. von (1895). Zu Riemann’s Abhandlung ‘Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse’,J. reine ang. Math. 114, 255–305.

    MATH  Google Scholar 

  • Migotti, A. (1883). Zur Theorie der Kreisteilungsgleichung, Z. B. der Math.-Naturwiss.Classe der Kaiserlichen Akademie der Wissenschaften, Wien, 87, 7–14.

    Google Scholar 

  • Montgomery, H. L., and Vaughan, R. C. (1975). The exceptional set in Goldbach’s problem,Acta Arith. 27, 353–370.

    MATH  MathSciNet  Google Scholar 

  • Schur, I. (c. 1935). Letter to Landau; see E. Lehmer (1936).

  • Vallée Poussin, C.-J. de la (1896). Recherches analytiques sur la théorie des nombres; Première partie: La fonction ζ(s) de Riemann et les nombres premiers en général,Ann. Soc. Sci. Bruxelles, 20, 183–256.

    Google Scholar 

  • Vaughan, R. C. (1977). Semmes trigonométriques sur les nombres premiers,C.R. Acad. Sci., Paris, A, 285, 981–983.

    MATH  MathSciNet  Google Scholar 

  • Vaughan, R. C. (1981).The Hardy-Littlewood method, Cambridge University Press.

  • Vinogradov, I. M. (1937). Some theorems concerning the theory of primes,Rec. Math. 2 (44), 179–195.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Imperial College of Science and Technology, Queen’s Gate, SW7 2BZ, London, England

    R. C. Vaughan

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  1. R. C. Vaughan
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This article is taken from an inaugural lecture delivered at Imperial College, London, on November 20, 1984.

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Vaughan, R.C. Adventures in arithmetick, or: How to make good use of a fourier transform. The Mathematical Intelligencer 9, 53–60 (1987). https://doi.org/10.1007/BF03025900

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