Additive functions in short intervals, gaps and a conjecture of Erdős
Bibliographic record
Abstract
Abstract With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki–Radziwiłł theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomäki–Radziwiłł theorem for divisor-bounded multiplicative functions recently proven in Mangerel (Divisor-bounded multiplicative functions in short intervals. arXiv: 2108.11401 ). We consider two sets of applications of these methods. Our first application shows that for an additive function $${\varvec{g:}} \mathbb {N} \rightarrow \mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>:</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> any non-trivial savings in the size of the average gap $$|{\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}}-{\varvec{g}}{} {\textbf {(}}{\varvec{n}}-{\textbf {1}}{} {\textbf {)}} |$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>)</mml:mo> <mml:mo>-</mml:mo> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mrow /> <mml:mo>)</mml:mo> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> implies that $${\varvec{g}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> must have a small first centred moment i.e. the discrepancy of $${\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdős relating to characterizing constant multiples of $${{\textbf {log}}} \,{\varvec{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>log</mml:mi> <mml:mspace /> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:mrow> </mml:math> as the only almost everywhere increasing additive functions. We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well approximated by a constant times a logarithm. We also show that if the set $$\{{\varvec{n}} \in \mathbb {N} : {\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}} < {\varvec{g}}{} {\textbf {(}}{\varvec{n}}-{\textbf {1}}{} {\textbf {)}}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>)</mml:mo> <mml:mo><</mml:mo> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> <mml:mrow /> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mrow /> <mml:mo>)</mml:mo> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is sufficiently sparse, and if $${\varvec{g}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> is not extremely large too often on the primes (in a precise sense), then $${\varvec{g}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> is identically equal to a constant times a logarithm.
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How this classification was reachedexpand
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 | 0.002 | 0.000 |
| 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 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".