Abstract
We connect a primitive operation from arithmetic—summing the digits of a base-B integer—to q-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of “B-ary” Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series.
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Notes
For example, summing both sides of [10, Theorem 3] over \(0\le k\le N\) and letting \({N\rightarrow \infty }\) gives a digit sum proof of the identity \(\prod _{n=1}^{\infty }(1-q^n)\sum _{k=0}^{\infty }{q^{k(k-1)/2}}{\prod _{i=1}^{k}(1-q^i)^{-1}}=\prod _{n=0}^{\infty }(1+q^n)\).
This formula looks like an extended version of identities used in [9].
A variant of \(L_2(q)\) is studied in [14].
This two-variable function transforms just like (13) under \(q\mapsto q^{B^j}\).
Without the multiplicative factor, \(L_B(q)\) generates the carry sum \(c_B(n-1,1)={\widehat{c}}_B(n-1,1)/(B-1)\).
This is an analog of the classical sum of divisors function\(\sigma (n)\). Note \(S_B(n)\) is equal to \(\sigma (n)\) if \(n<B\), and generally, the difference between the functions (by Corollary 4) is \(\sigma (n)-S_B(n)=\sum _{d|n}{\widehat{c}}_B(\{ 1\}^d)\).
E.g., see [7, p. 300, Prob. 11].
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Acknowledgements
We wish to thank the anonymous referees for useful suggestions that strengthened this work. The first author was partially funded by HCSSiM at Hampshire College during the preparation of this paper; special thanks to Prof. David Kelly.
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Schneider, M., Schneider, R. Digit sums and generating functions. Ramanujan J 52, 291–302 (2020). https://doi.org/10.1007/s11139-019-00193-6
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DOI: https://doi.org/10.1007/s11139-019-00193-6