Partition Theory and Modular Forms
Summary
Partition theory studies the ways in which a positive integer can be expressed as a sum of positive integers, without regard to order. Originating in the work of Euler, it has evolved through the discovery of generating functions—formal power series whose coefficients enumerate partitions—and celebrated results such as the Pentagonal Number Theorem. Modular forms arise in this context when these generating functions exhibit specific transformation properties under the modular group, revealing deep connections between combinatorics, complex analysis and number theory. The Dedekind eta function, a prototypical modular form given by an infinite product, serves as a building block for many partition‐related q-series. In recent decades, generalisations such as mock modular forms and weak Maass forms have emerged, driven by the need to explain Ramanujan’s enigmatic mock theta functions and to uncover congruence phenomena. These structures underpin advances ranging from asymptotic formulas for partition growth to applications in mathematical physics and representation theory.
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Partition Theory and Modular Forms publication trend
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Technical terms
Partition: A representation of a positive integer as a sum of positive integers, order disregarded.
Generating function: A formal power series whose coefficient of qⁿ counts the number of partitions of n.
q-series: A power series in the variable q, often serving as a generating function in partition theory.
Modular form: A complex analytic function on the upper half-plane that transforms in a prescribed way under the action of the modular group.
Dedekind eta function: An infinite-product modular form essential in constructing partition generating functions.
Weak Maass form: A generalisation of modular forms allowing non-holomorphic terms, instrumental in modern congruence proofs.
References
- On a Tauberian theorem of Ingham and Euler–Maclaurin summation. The Ramanujan Journal (2021).
- Congruences for Andrews’ spt-function modulo powers of 5 5 , 7 7 and 13 13. Transactions of the American Mathematical Society (2012).
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