Abstract
Let J(N, H) be the Selberg integral and E(x, T) the error term in Kaczorowski-Perelli's weighted form of the classical explicit formula. We prove that the estimate J(N, H) = o(H2 N) is connected with an appropriate estimate of ∫N 2N| E(x,T)2 dx, uniformly for H and T in some ranges. Moreover, assuming a suitable bound for ∫N 2N| E(x,T)|2 dx, we also obtain, for all sufficiently large N and H ≫ (log N)11/12, that every interval [N,N + H] contains ≫ H Goldbach numbers.
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References
R. Baker, G. Harman and J. Pintz, The exceptional set for Goldbach's problem in short intervals in Sieve Methods, Exponential Sums and their Applications in Number Theory (ed. G. R. H. Greaves et al.) Cambridge U. P. (1996), pp. 1–54.
J. B. Conrey, At least two fifths of the zeros of the Riemann zeta function are on the critical line, Bull. A.M.S. 20 (1989), 79–81.
G. Coppola and A. Vitolo, On the distribution of primes in intervals of length logθ N, Acta Math. Hungar. 70 (1996), 151–166.
D. A. Goldston, Linnik's theorem on Goldbach numbers in short intervals, Glasgow Math. J. 32 (1990), 285–297.
D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals, in Analytic Number Theory and Dioph. Probl. (ed. by A. C. Adolphson et al.), Birkhäuser Verlag (1987), pp. 183–203.
D. R. Heath-Brown, The differences between consecutive primes, J. London Math. Soc. 18 (1978), 7–13.
A. Ivié, The Riemann Zeta-function J. Wiley (1985).
C. Jia, Almost all short intervals contain prime numbers, Acta Arith. 76 (1996), 21–84.
J. Kaczorowski and A. Perelli, A new form of the Riemann-von Mangoldt explicit formula, Boll. U.M.I. 10-B (1996), 51–66.
J. Kaczorowski and A. Perelli, On the distribution of primes in short intervals, J. Math. Soc. Japan 45 (1993), 447–458.
I. Kátai, A remark on a paper of Ju. V. Linnik, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl 17 (1967), 99–100 (in Hungarian).
A. Languasco and A. Perelli, On Linnik's theorem on Goldbach numbers in short intervals and related problems, Ann. Inst. Fourier 44 (1994), 307–322.
H. L. Montgomery, Topics in Multiplicative Number Theory Springer L.N. 227 (1971).
H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353–370.
A. Selberg, On the normal density of primen in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87–105.
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Languasco, A. A Note on Primes and Goldbach Numbers in Short Intervals. Acta Mathematica Hungarica 79, 191–206 (1998). https://doi.org/10.1023/A:1006553707162
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DOI: https://doi.org/10.1023/A:1006553707162