Generating Special Arithmetic Functions by Lambert Series Factorizations




We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing  results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and conjectures related to the inverse matrix entries defined in Schmidt's article and the Euler partition function $p(n)$ which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article.

Author Biography

Maxie Dion Schmidt, Georgia Institute of Technology

I am currently a first-year Ph.D. student in the School of Mathematics at Georgia Institute of Technology. I have a MS in Computer Science from the University of Iliinois at Urbana-Champaign. Please see my website ( for more information about publications, awards, and honors I have received.


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\newblock Oxford University Press, 2008.

M. Merca, {Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer},
{\it J. of Number Theor.}, 160, pp. 60--75 (2016).

M. Merca, {The {L}ambert series factorization theorem},
{\it Ramanujan J.}, pp. 1--19 (2017).

F.~W.~J. Olver, D.~W. Lozier, R.~F. Boisvert, and C.~W. Clark.
\newblock {\it {NIST} Handbook of Mathematical Functions}.
\newblock Cambridge University Press, 2010.

M. D. Schmidt, {Combinatorial sums and identities involving generalized
divisor functions with bounded divisors}, 2017,

M. D. Schmidt, {New recurrence relations and matrix equations for arithmetic
functions generated by {L}ambert series}, 2017,
Tentatively accepted in \textit{Acta Arith.}

N. J. A. Sloane, {The Online Encyclopedia of Integer Sequences}, 2017,