The Riemann zeta-function ζ(\ud835\udc60) for \ud835\udc60 ∈ \u2102, is a complex function fundamental to prime number theory. Computation of ζ(\ud835\udc60) along the critical line \ud835\udc60 = 1/2 + I\ud835\udc61, \ud835\udc61 ∈ \u211d, is important for the distribution of primes, since all the zeta-zeros are predicted to lie here (the Riemann Hypothesis). Hardy’s Z-function \ud835\udc4d (\ud835\udc61) = \ud835\udc52I\ud835\udf03 \ud835\udc61 ζ(1/2 + I\ud835\udc61 ), \ud835\udf03 (\ud835\udc61) ∈ \u211d , defines the amplitude of the zeta-function along the critical line. Such computations also provide a means of Riemann Hypothesis verification, within closed intervals, and estimates on the bounds of ζ(1/2 + I\ud835\udc61) , central to the Lindelöf Hypothesis.
\nUntil recently, the most efficient means of computing \ud835\udc4d (\ud835\udc61) employed the Riemann-Siegel (RS) formula, an \ud835\udc42 (√ \ud835\udc61) operational method. In 2011, this was superseded by an algorithm devised by G. A. Hiary, requiring just \ud835\udc42 (\ud835\udc611/3{\ud835\udc59\ud835\udc5c\ud835\udc54 \ud835\udc61 }\ud835\udf05 )operations. The methodology involved the sub-division of the RS formula into sequences of quadratic Gauss sums of length \ud835\udc41. Such sums are amenable to rapid computation, in \ud835\udc42(\ud835\udc59\ud835\udc5c\ud835\udc54(\ud835\udc41)) operations, using a recursive scheme. Central to this work is a new asymptotic formula for \ud835\udc4d(\ud835\udc61), with some interesting analytical properties. Computationally, this new formulation allows one to improve on the quadratic sum expansion and express \ud835\udc4d(\ud835\udc61) in terms of sub-sequences of \ud835\udc5a\ud835\udc61\u210e-order Gauss sums. In the cubic \ud835\udc5a = 3 case,these sums can be computed rapidly, utilising a similar scheme to that used for the quadratics. The result is a more efficient algorithm, requiring only \ud835\udc42 ((\ud835\udc61 /\ud835\udf00\ud835\udc61)1/4{ \ud835\udc59\ud835\udc5c\ud835\udc54 (\ud835\udc61) }3) operations. Sample computations, using the open-source code developed for this eCSE Project and available on GitHub repository, support these findings.
", "languages": [ { "id": "eng", "title": { "en": "English" } } ], "publication_date": "2026-03-19", "publisher": "Zenodo", "resource_type": { "id": "poster", "title": { "de": "Poster", "en": "Poster" } }, "rights": [ { "description": { "en": "" }, "icon": "cc-by-nc-nd-icon", "id": "cc-by-nc-nd-4.0", "props": { "scheme": "spdx", "url": "https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode" }, "title": { "en": "Creative Commons Attribution Non Commercial No Derivatives 4.0 International" } } ], "subjects": [ { "subject": "Riemann zeta-function" }, { "subject": "Hardy function" }, { "subject": "Gauss sums" }, { "subject": "Operational optimisation" }, { "subject": "ARCHER2" }, { "subject": "eCSE" }, { "subject": "Celebration of Science" } ], "title": "New, high-performance algorithms for the computation of the Hardy Function", "version": "1.0" }, "parent": { "access": { "owned_by": { "user": "48291" }, "settings": { "accept_conditions_text": null, "allow_guest_requests": false, "allow_user_requests": false, "secret_link_expiration": 0 } }, "communities": { "default": "ac3b180d-8b5d-4f22-84a9-35adfb25eed3", "entries": [ { "access": { "member_policy": "open", "members_visibility": "public", "record_submission_policy": "open", "review_policy": "open", "visibility": "public" }, "children": { "allow": false }, "created": "2021-03-10T16:01:47.749254+00:00", "custom_fields": {}, "deletion_status": { "is_deleted": false, "status": "P" }, "id": "ac3b180d-8b5d-4f22-84a9-35adfb25eed3", "links": {}, "metadata": { "curation_policy": "This community is to share the open access materials relevant to ARCHER2 only.
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