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Jun 13 |
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Quasicrystals and the Riemann Hypothesis
There are two papers of Bombieri where the relevance of the explicit formula to the truth of the Riemann Hypothesis is investigated. Here they are: "A variational approach to the explicit formula", Comm. Pure Appl. Math. 56 (2003), no. 8, 1151–1164. and "Remarks on Weil's quadratic functional in the theory of prime numbers. I.", Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 3, 183–233 (2001). |
Jun 13 |
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Quasicrystals and the Riemann Hypothesis
... I need to think more about it but certainly most versions of the explicit formula that I came accross assumes that at least one of $h$ or $\widehat{h}$ is analytic in some strip, since complex analysis is used in the proof. The only exception is arxiv.org/abs/1203.5328 (section 2) where the explicit formula is proven by a rather convoluted method, but it might look a bit more like what you would like to do... |
Jun 13 |
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Quasicrystals and the Riemann Hypothesis
The way I see it your equation above is a suggestive and heuristic way of writting the explicit formula. Certainly if you integrate against certain smooth functions you will recover (more or less) the explicit formula (some terms seems to be missing in the formula above). However I don't think that the explicit formula is true for any smooth test function, and in particular this might suggest that the distributional formula above is not exactly "true 100% of the time"... |
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Jun 13 |
answered | Quasicrystals and the Riemann Hypothesis |