Modular forms arising from divisor functions

DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES

We consider Weierstrass functions and divisor functions arising from q-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. ...

Modular Divisor Functions and Quadratic Reciprocity

Modular Functions and Modular Forms

These are the notes for Math 678, University of Michigan, Fall 1990, exactly as they were handed out during the course except for some minor revisions and corrections. Please send comments and corrections to me at [email protected].

CLASSICAL AND p-ADIC MODULAR FORMS ARISING FROM THE BORCHERDS EXPONENTS OF OTHER MODULAR FORMS

Abstract. Let f(z) = q ∏∞ n=1(1−q) be a modular form on SL2(Z). Formal logarithmic differentiation of f yields a q-series g(z) := h −∞n=1 ∑ d|n c(d)dq n whose coefficients are uniquely determined by the exponents of the original form. We provide a formula, due to Bruinier, Kohnen, and Ono for g(z) in terms of the values of the classical j-function at the zeros and poles of f(z). Further, we giv...

MODULAR FORMS ARISING FROM Q(n) AND DYSON’S RANK

Abstract. Let R(w; q) be Dyson’s generating function for partition ranks. For roots of unity ζ 6= 1, it is known that R(ζ; q) and R(ζ; 1/q) are given by harmonic Maass forms, Eichler integrals, and modular units. We show that modular forms arise from G(w; q), the generating function for ranks of partitions into distinct parts, in a similar way. If D(w; q) := (1 + w)G(w; q) + (1− w)G(−w; q), the...