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Sums of three or more primes. (English) Zbl 1036.11506
Summary: It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error \(\sum_{p \leq x} \log p - x\) in the Prime Number Theorem, such bounds being within a factor of \((\log x)^{2}\) of each other and this fact being equivalent to the Riemann Hypothesis. In this paper the authors show that, provided “Riemann Hypothesis” is replaced by “Generalized Riemann Hypothesis”, results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of \(k\) primes for \(k \geq 4\), and, in a mean square sense, for \(k \geq 3\). We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a “Quasi-Riemann Hypothesis”. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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[1] R. C. Baker and G. Harman, Diophantine approximation by prime numbers, J. London Math. Soc. (2) 25 (1982), no. 2, 201 – 215. · Zbl 0443.10015
[2] Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. · Zbl 0453.10002
[3] D. A. Goldston, On Hardy and Littlewood’s contribution to the Goldbach conjecture, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) Univ. Salerno, Salerno, 1992, pp. 115 – 155. · Zbl 0792.11039
[4] G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio Numerorum’: III On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70. · JFM 48.0143.04
[5] D. R. Heath-Brown, Zero density estimates for the Riemann zeta-function and Dirichlet \?-functions, J. London Math. Soc. (2) 19 (1979), no. 2, 221 – 232. · Zbl 0393.10043
[6] M. N. Huxley and H. Iwaniec, Bombieri’s theorem in short intervals, Mathematika 22 (1975), no. 2, 188 – 194. · Zbl 0317.10048
[7] M. N. Huxley and M. Jutila, Large values of Dirichlet polynomials. IV, Acta Arith. 32 (1977), no. 3, 297 – 312. · Zbl 0352.10019
[8] A. E. Ingham, Some asymptotic formulae in the theory of numbers, J. London Math. Soc. 2 (1927), 202-208. · JFM 53.0157.01
[9] Yu. V. Linnik, A new proof of the Goldbach-Vinogradov theorem, Mat. Sb. 61 (1946), 3-8. · Zbl 0063.03589
[10] Hugh L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. · Zbl 0216.03501
[11] H. L. Montgomery and R. C. Vaughan, Error terms in additive prime number theory, Quart. J. Math. Oxford Ser. (2) 24 (1973), 207 – 216. · Zbl 0257.10027
[12] A. E. Özlük , Pair correlation of zeros of Dirichlet \(L\)-functions Thesis, University of Michigan, 1982.
[13] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. · Zbl 0601.10026
[14] Robert-C. Vaughan, Sommes trigonométriques sur les nombres premiers, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A981 – A983 (French, with English summary). · Zbl 0374.10025
[15] R. C. Vaughan, The Hardy-Littlewood method, Cambridge Tracts in Mathematics, vol. 80, Cambridge University Press, Cambridge-New York, 1981. · Zbl 0455.10034
[16] I. M. Vinogradov, Representation of an odd number as a sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937), 6-7.
[17] C. Yalçin Yildirim, The pair correlation of zeros of Dirichlet \?-functions and primes in arithmetic progressions, Manuscripta Math. 72 (1991), no. 3, 325 – 334. · Zbl 0734.11045
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