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Bell and Smertnig (2026): Mahler Series with Multiplicative Coefficients
theorem_1_3: In characteristic zero, multiplicative coefficients of a k-Mahler series are k-regular and split into a recurrence on powers of one prime and a power times an eventually periodic factor on coprime indices.
Jason Bell and Daniel Smertnig, Mahler series with multiplicative coefficient sequences, arXiv:2603.23456v1, submitted 24 March 2026. The selected source is this preprint version; no journal acceptance or later correction is asserted.
Source identity and result
The copy read for this card is the 29-page version-one manuscript. Its header is PDF 1.7 and its size is 687642 bytes. The version and submission date were checked against the arXiv record on 17 September 2026. Printed and PDF page numbers agree. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2603.23456), every other right reserved.
Theorem 1.3, p. 2, gives regularity and an explicit prime-power decomposition for multiplicative coefficients of a Mahler series in characteristic zero. Definitions are on pp. 4–6; the proof is assembled on p. 27 from the regular-sequence classification in Section 3 and the two Mahler-to-regular cases in Sections 5 and 7.
Corollary 1.4, p. 3, omits multiplicativity in its printed statement, although its proof on p. 27 uses Theorem 1.3. This record consumes the explicitly stated Theorem 1.3 with its multiplicativity hypothesis. It does not adopt the printed corollary without that qualification.
Relevance and reading limits
For the totient coefficients, the decomposition would force an eventually periodic function to take infinitely many distinct values on primes. The resulting non-Mahler conclusion is a functional statement.
The source has been read in full as extracted text. This extraction checks the statement against the p. 2 image and reads the definitions on pp. 4–6, the totient example on p. 3, and the proof assembly on p. 27. It provides a statement and a proof pointer, not a complete source-proof reconstruction or independent acceptance.
The imported automatic-sequence classification, reduction/lifting result, rational multiplicative-sequence classification and Mahler-denominator theorems remain external premises. Their original proofs have not been read for this filing. Complete proof compilation and its independent review remain outstanding; no native verification tier is assigned.
Bears on. Problem 249: this excludes a finite Mahler equation for its generating function, while leaving the irrationality of its value at one half unresolved.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.