Computing the truncated theta function via Mordell integral
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Hiary has presented an algorithm which allows us to evaluate the truncated theta function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 0 Overscript n Endscripts exp left-parenthesis 2 pi normal i left-parenthesis z k plus tau k squared right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:mi>exp</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mi> τ </mml:mi> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{k=0}^n \exp (2\pi \mathrm {i} (zk+\tau k^2))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to within <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus epsilon"> <mml:semantics> <mml:mrow> <mml:mo> ± </mml:mo> <mml:mi> ϵ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\pm \epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis ln left-parenthesis StartFraction n Over epsilon EndFraction right-parenthesis Superscript kappa Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ln</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mi> ϵ </mml:mi> </mml:mfrac> </mml:mstyle> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> κ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(\ln (\tfrac {n}{\epsilon })^{\kappa })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> arithmetic operations for any real <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z"> <mml:semantics> <mml:mi>z</mml:mi> <mml:annotation encoding="application/x-tex">z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau"> <mml:semantics> <mml:mi> τ </mml:mi> <mml:annotation encoding="application/x-tex">\tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This remarkable result has many applications in Number Theory, in particular, it is the crucial element in Hiary’s algorithm for computing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta left-parenthesis one half plus normal i t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ζ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\zeta (\tfrac {1}{2}+\mathrm {i} t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to within <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus t Superscript negative lamda"> <mml:semantics> <mml:mrow> <mml:mo> ± </mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi> λ </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\pm t^{-\lambda }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O Subscript lamda Baseline left-parenthesis t Superscript one third Baseline ln left-parenthesis t right-parenthesis Superscript kappa Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> λ </mml:mi> </mml:mrow>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it