The classical absolute-value conjecture for Xi zeroes

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Let Ξ(χ,t)=Λ(χ,1/2+it)\Xi(\chi,t)=\Lambda(\chi,1/2+it) for an abelian character χ\chi over a number field. The classical absolute-value conjecture. For every e≥0e\geq0, there are at most two zeroes of Ξ(χ,t)\Xi(\chi,t), counted without multiplicity, having absolute value ee.

This is presented as the direct classical analogue of the characteristic-pp reciprocal-zero conjecture. The bound of two reflects the functional equation sending tt to −t-t, while the conjecture remains unproved.

References

Primary source

David Goss, “A Riemann Hypothesis for characteristic p L-functions”, arXiv:math/9907019 (1999).

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