Manchester Number Theory Seminar - Daniel Shiu (Manchester)
| Dates: | 27 October 2026 |
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| Times: | 15:00 - 15:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Daniel Shiu (Manchester)
Title: Admissible sets of integers in imaginary quadratic fields of class number 1
Abstract: Let O_K be the ring of integers of a number field K. A finite subset H={eta_1,...,eta_k} of O_K is said to be admissible if for every prime ideal P in O_K, the reductions of the eta_i modulo P do not represent every residue class.
Admissible sets are of interest as shifts of moderate-sized admissible sets have been shown to capture multiple primes (Castillo et al after Zhang/Maynard/Tao/Polymath) and the existence of dense admissible sets of rational integers demonstrates the incompatibility of the first and second Hardy-Littlewood conjectures (Hensley and Richards).
We construct dense admissible sets of integers in the quadratic fields of class number 1 (Gaussian integers, Eisenstein integers etc.). We find that methods for constructing dense admissible sets of rational integers do not transfer well. Focusing on the case of sets of size 50, we are able to produce new records for the shortest norm of the difference between large primes in these fields. Focusing on discs with more admissible elements than the prime count when the disc is shifted to the origin, we are able to construct explicit counter-examples to an extension of the second Hardy-Littlewood conjecture in some of these fields.
Room: Frank Adams 1, Alan Turing Building
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Frank Adams 1
Alan Turing Building
Manchester
