A Note on Primes and Goldbach Numbers in Short Intervals |
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Authors: | A Languasco |
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Institution: | (1) Dipartimento di Matematica, Università di Genova Via Dodecaneso, 35 16146 Genova, Italy |
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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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Keywords: | |
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