Related papers: Euler-like recurrences for smallest parts function…
A theorem due to D. Bernstein states that Euler characteristic of a hypersurface defined by a polynomial f in (C\{0})^n is equal (upto a sign) to n! times volume of the Newton polyhedron of f. This result is related to algebaric torus…
We describe a complete algorithm to compute millions of coefficients of classical modular forms in a few seconds. We also review operations on Euler products and illustrate our methods with a computation of triple product L-function of…
This note rederives a formula for $r$-color partitions, $1 \le r \le 24$, including Rademacher's celebrated result for ordinary partitions, from the duality between modular forms of weights $-r/2$ and $2+r/2$.
Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate…
In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of…
We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…
In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an…
We provide a natural interpretation of the secondary Euler characteristic and introduce higher Euler characteristics. For a compact oriented manifold of odd dimension, the secondary Euler characteristic recovers the Kervaire…
We prove multiplicative congruences mod $2^{12}$ for George Andrews's partition function, $\overline{\mathcal{EO}}(n)$, the number of partitions of $n$ in which every even part is less than each odd part and only the largest even part…
In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the…
We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We…
In this semi-expository note, we give a new proof of a structure theorem due to Shimura for nearly holomorphic modular forms on the complex upper half plane. Roughly speaking, the theorem says that the space of all nearly holomorphic…
New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a normalized…
Aiming at a complete classification of unitary N=2 minimal models (where the assumption of space-time supersymmetry has been dropped), it is shown that each modular invariant candidate of a partition function for such a theory is indeed the…
In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke…
Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions…
This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.
The present paper is a continuation of our work [11], where we introduced a fractional operator calculus related to a fractional ${\psi}-$Fueter operator in the one-dimensional Riemann-Liouville derivative sense in each direction of the…
We obtain a new family of relations satisfied by the partition function. In contrast with most partition relations, these involve non-trivial roots of unity. We present two proofs, one using the fact that the discriminant modular form is a…
We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then…