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MacMahon's sums-of-divisors and allied q-series
... One can argue that the main merit of our effort here lies in inviting the audience to a variety to the techniques employed for the present goal, that the authors believe should help in similar circumstances. The first of such installments appeared in [2] for the q-series U 2t (q): Theorem 1.1. If t is a non-negative integer, then we have that ...
... In the same spirit but in an earlier paper [2], the first and the third authors together with K. Ono have found such an explicit description for MacMahon's quasimodular form ...
... An important observation is that Ramanujan's U 2t (q) and MacMahon's U 2t (q) are directly linked to each other. In fact, this fact allows us to formulate Theorem 1.3 of [2] as follows: ...
In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite families of q-series by Ramanujan. Our main emphasis will be on explicit representations for a variety of q-series, studied primarily by MacMahon and Ramanujan, with an eye towards their modular properties and their proper place in the ring of quasimodular forms of level one and level two.
... Modular forms and quasimodular forms are foundational objects in number theory; see the texts [23,27,30,56,66] for many examples. The quasimodularity of MacMahon's series has lead to much study from the number-theoretic perspective in recent years [1,4,9,12,15,18,26,28,44,51,54,57,58,68]. In particular, it is easy to show that the Fourier coefficients of U a (q) can be expressed as very natural sums of products of the multiplicities of certain classes of partitions. ...
... For specifics of their work, we refer the reader to Section 3.1. These number-theoretic connections are deep and have, for instance, produced connections between q-multiple zeta values with weakly holomorphic modular forms [44,57], partition congruences [4], higher Appell functions [18], and prime numbers [28]. In many of these connections, it is natural from the number-theoretic point of view to interpret the multiple divisor sums as sums over partitions into a fixed number of part sizes; thus this theory achieves a unification of q-multiple zeta values, partitions, and quasimodular forms. ...
... As usual we have z = z 1 + z 2 i ∈ H. The series (2.1) converges for Re(s) + ℓ > 2 by the classical Hecke trick, and continues meromorphically to all s ∈ C such that 4 ...
In recent years, the generalized sum-of-divisor functions of MacMahon have been unified into the algebraic framework of q-multiple zeta values. In particular, these results link partition theory, quasimodular forms, q-multiple zeta values, and quasi-shuffle algebras. In this paper, we complete this idea of unification for higher levels, demonstrating that any quasimodular form of weight and level N may be expressed in terms of the q-multiple zeta values of level N studied algebraically by Yuan and Zhao. We also give results restricted to q-multiple zeta values with integer coefficients, and we construct completely additive generating sets for spaces of quasimodular forms and for quasimodular forms with integer coefficients. We also provide a variety of computational examples from number-theoretic perspectives that suggest many new applications of the algebraic structure of q-multiple zeta values to quasimodular forms and partitions.
... The function U t (−2, q) has received much attention in recent years; see [1,3,5,16] for more details about the results that have already been proven. ...
... Next, we focus our attention on past results which already appear in the literature. The case a = −2 has been investigated in [1] and [3]. In [1], Amdeberhan, Andrews, and Tauraso discovered that, for all N ≥ 0, ...
In this paper, we prove several new infinite families of Ramanujan--like congruences satisfied by the coefficients of the generating function which is an extension of MacMahon's generalized sum-of-divisors function. As a by-product, we also show that, for all , where is the number of almost 3-regular overpartitions of n.
... The Eisenstein series E 2k (q) corresponds to the partition λ = (k), as we have E (k 1 ) (q) = E 2k (q) 1 . ...
... Such traces arise in recent work on MacMahon's sums-of-divisors q-series (see Thm. 1.4 of [1]). For partitions λ = (1 m 1 , . . . ...
In his ``lost notebook'', Ramanujan used iterated derivatives of two theta functions to define sequences of q-series and that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of ``partition Eisenstein series'', extensions of the classical Eisenstein series defined by For functions \phi : \mathcal{P}\mapsto \C on partitions, the {\it weight 2n partition Eisenstein trace} is \Tr_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). For all t, we prove that U_{2t}(q)=\Tr_t(\phi_u;q) and V_{2t}(q)=\Tr_t(\phi_v;q), where and are natural partition weights, giving the first explicit quasimodular formulas for these series.
... Remark. Such traces arise in recent work on MacMahon's sums-of-divisors q-series (see Theorem 1.4 of [1]) and in an explicit proof of a claim from Ramanujan's "lost notebook" on q-series assembled from derivatives of theta functions (see Theorem 1.2 of [2]). ...
We study "partition Eisenstein series", extensions of the Eisenstein series defined by For functions on partitions, the weight 2k "partition Eisenstein trace" is the quasimodular form These traces give explicit formulas for some well-known generating functions, such as the kth elementary symmetric functions of the inverse points of 2-dimensional complex lattices as well as the 2kth power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.
We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n ≥ 2 is prime if and only if ( 3 n 3 − 13 n 2 + 18 n − 8 ) M 1 ( n ) + ( 12 n 2 − 120 n + 212 ) M 2 ( n ) − 960 M 3 ( n ) = 0 ,
where the M a ( n ) are MacMahon’s well-studied partition functions. More generally, for MacMahonesque partition functions M a → ( n ) , we prove that there are infinitely many such prime detecting equations with constant coefficients, such as 80 M ( 1 , 1 , 1 ) ( n ) − 12 M ( 2 , 0 , 1 ) ( n ) + 12 M ( 2 , 1 , 0 ) ( n ) + ⋯ − 12 M ( 1 , 3 ) ( n ) − 39 M ( 3 , 1 ) ( n ) = 0 .
In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so, we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions. One such example involves multiple q-harmonic sums ∑k=1n(-1)k-1nkq(1+qk)qk2+tk[k]q2tn+kkq=∑1≤k1≤⋯≤k2t≤nqn+k1+k3⋯+k2t-1+qk2+k4+⋯+k2t[n+k1]q[k2]q⋯[k2t]q.
In 1919, P. A. MacMahon studied generating functions for generalized divisor
sums. In this paper, we provide a framework in which to view these generating
functions in terms of Jacobi forms, and prove that they are quasi-modular
forms.
New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their general form by relating them to quasi-modular forms. These functions arise as solutions to a curve-counting problem on Abelian surfaces. Comment: 6 Pages
We study \(
\mathcal{N} = 2
\) supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in the Ω-background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.
These representations allow us to derive rigorously the Seiberg-Witten geometry, the curves, the differentials, and the prepotential.
We study pure 525-03 = 2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five-dimensional theory compactified on a circle.
In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \(
{\tilde M_*}({\Gamma _1})
\) denote the graded ring of quasi-modular forms on the full modular group Γ= PSL(2, ℤ). This is the ring generated by G2, G4, G6, and graded by assigning to each Gk
the weight where \(
{G_k} = - \frac{{{B_k}}}{{2k}} + \sum\limits_{n = 1}^\infty {\left( {{{\sum\limits_{d|n} d }^{k - 1}}} \right)} {q^n}\left( {k \geqslant 2,{B_k} = kth Bernoulli number} \right)
\) are the classical Eisenstein series, all of which except G
2 are modular.
In this article, we use the triple product identity and the quintuple product identity to derive Ramanujan's famous differential equations for the Eisenstein series.
Denote by
the Euler product, and by
the partition generating function. More generally, if k is any integer, put
so that p(n) = p −1(n) . In [3], Atkin proved the following theorem.
The work I shall describe in these lectures has two themes, a classical one going back to Ramanujan [8] and a modern one initiated by Serre [9] and Deligne [3]. To describe the classical theme, let the unique cusp form of weight 12 for the full modular group be written
\Delta = q\mathop \prod \limits_1^\infty (1 - q^n )^{24} = \sum\limits_1^\infty {\tau (n)q^n }
(1) and note that the associated Dirichlet series has an Euler product (2) so that all the τ(n) are known as soon as the τ(p) are.
Soient K un corps de nombres algébriques totalement réel, et ζK sa fonction zêta. D’après un théorème de Siegel [24], ζK(1 − k) est un nombre rationnel si k est entier ⩾ 1; il est ≠ 0 si k est pair. Lorsque K est abélien sur Q, on peut écrire ce nombre comme produit de «nombres de Bernoulli généralisés»:
\zeta _K (1 - k) = \mathop \prod \limits_X L(X,1 - k) = \mathop \prod \limits_X ({{ - b_k (X)} \mathord{\left/
{\vphantom {{ - b_k (X)} k}} \right.
\kern-\nulldelimiterspace} k}), cf. [18],
, où χ parcourt l’ensemble des caractères de Q attachés à K. Cela permet de démontrer des propriétés de congruence reliant les ζK(1−k) pour diverses valeurs de k, et d’en déduire par interpolation une fonction zêta p-adique pour le corps k, au sens de Kubota-Leopoldt (cf. [7], [10], [11], [16]).
Andrews' spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only know previously for sufficiently large n. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived. Comment: 19 pages, added extra section for concluding remarks, corrected some typos, added more acknowledgments, added urls to references
The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of t-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores. Several applications are derived, including the "marked hook formula".
Explicit formulas for MacMahon's Ar(q) and Cr(q), (e-mail
- H Bachmann