The 2-Adic Valuation of Spectral Bernoulli Numbers: A Digit-Sum Cancellation Law
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2026
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| author | EL MAHYAOUI, Nabil |
| author_facet | EL MAHYAOUI, Nabil |
| contents | <p>For n = 2<sup>r</sup> with r ≥ 1, the spectral Bernoulli numbers B<sub>m</sub><sup>(n)</sup> defined through the logarithmic derivative of W<sub>n</sub> = ∏<sub>k=0</sub><sup>n−1</sup> E<sub>n</sub><sup>k</sup> satisfy</p> <p>v<sub>2</sub>(B<sub>m</sub><sup>(n)</sup>) = 4m + 2r − 3   for all m ≥ 1.</p> <p>The 2-adic valuation is <strong>exactly linear</strong> in m with universal slope 4 and intercept 2r − 3, independent of the binary expansion of m.</p> <p>This uniformity arises from a <strong>digit-sum cancellation</strong>: the factorial v<sub>2</sub>((nm)!) and the Taylor coefficient v<sub>2</sub>(b<sub>m</sub>) each depend on the binary digit sum s<sub>2</sub>(m), but with opposite signs, so the dependence cancels in the product B<sub>m</sub><sup>(n)</sup> = (nm)! · b<sub>m</sub>.</p> <p>We prove the closed form W<sub>4</sub>(x) = (sinh<sup>4</sup>x − sin<sup>4</sup>x)/16 and establish the valuation law via a Kummer-theoretic argument on the 2-adic inverse of a rescaled factorial series: the even-index step uses a direct parity argument, while the odd-index step employs an even/odd decomposition in which the dominant term is controlled by unshifted Kummer and a cross-convolution remainder is proved subdominant via a parity argument on carry-free decompositions.</p> <p>The general conjecture is verified computationally for n = 2, 4, 8, 16.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19168287 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The 2-Adic Valuation of Spectral Bernoulli Numbers: A Digit-Sum Cancellation Law EL MAHYAOUI, Nabil p-adic valuation spectral Bernoulli numbers digit sum 2-adic analysis Mittag-Leffler <p>For n = 2<sup>r</sup> with r ≥ 1, the spectral Bernoulli numbers B<sub>m</sub><sup>(n)</sup> defined through the logarithmic derivative of W<sub>n</sub> = ∏<sub>k=0</sub><sup>n−1</sup> E<sub>n</sub><sup>k</sup> satisfy</p> <p>v<sub>2</sub>(B<sub>m</sub><sup>(n)</sup>) = 4m + 2r − 3   for all m ≥ 1.</p> <p>The 2-adic valuation is <strong>exactly linear</strong> in m with universal slope 4 and intercept 2r − 3, independent of the binary expansion of m.</p> <p>This uniformity arises from a <strong>digit-sum cancellation</strong>: the factorial v<sub>2</sub>((nm)!) and the Taylor coefficient v<sub>2</sub>(b<sub>m</sub>) each depend on the binary digit sum s<sub>2</sub>(m), but with opposite signs, so the dependence cancels in the product B<sub>m</sub><sup>(n)</sup> = (nm)! · b<sub>m</sub>.</p> <p>We prove the closed form W<sub>4</sub>(x) = (sinh<sup>4</sup>x − sin<sup>4</sup>x)/16 and establish the valuation law via a Kummer-theoretic argument on the 2-adic inverse of a rescaled factorial series: the even-index step uses a direct parity argument, while the odd-index step employs an even/odd decomposition in which the dominant term is controlled by unshifted Kummer and a cross-convolution remainder is proved subdominant via a parity argument on carry-free decompositions.</p> <p>The general conjecture is verified computationally for n = 2, 4, 8, 16.</p> |
| title | The 2-Adic Valuation of Spectral Bernoulli Numbers: A Digit-Sum Cancellation Law |
| topic | p-adic valuation spectral Bernoulli numbers digit sum 2-adic analysis Mittag-Leffler |
| url | https://doi.org/10.5281/zenodo.19168287 |