Abstract
We show that, on the Riemann hypothesis, lim supX→∞ I(X)/X2 0.8603, where I(X) = 2X
X (ψ(x) − x)2 dx. This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that 1.86 · 10−4 I(X)/X2 for sufficiently large X, and that the I(X)/X2 has no limit as X → ∞.
X (ψ(x) − x)2 dx. This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that 1.86 · 10−4 I(X)/X2 for sufficiently large X, and that the I(X)/X2 has no limit as X → ∞.
Original language | English |
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Pages (from-to) | 740-762 |
Journal | Journal of Number Theory |
Volume | 238 |
Early online date | 26 May 2022 |
Publication status | Published - 1 Sept 2022 |
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Alam, S. R. (Manager), Williams, D. A. G. (Manager), Eccleston, P. E. (Manager) & Greene, D. (Manager)
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