A derivation of the Hardy-Ramanujan formula from an arithmetic formula

Abstract

Abstract. We re-prove the Hardy-Ramanujan asymptotic formula for the partition function without using the circle method. We derive our result from recent work of Bruinier and Ono on harmonic weak Maass forms.

Class invariants for certain non-holomorphic modular functions

“…Therefore, once one explicitly knows the higher level Heegner points, a similar analysis could be done to generate class invariants. For example, in level 6 for quadratic forms of discriminant −24n + 1 the Heegner points have been determined by Dewar and Murty, see [9]. In particular, Dewar and Murty's results were used in the paper of Mertens and Rolen ( [17]).…”

Algebraic and transcendental formulas for the smallest parts function

“…The authors [2] recovered (1.1) from Poincaré series representing a weight 5/2 harmonic Maass form whose shadow is η −1 (τ ). As pointed out by Bruinier and Ono [21], the exact formula can be recovered from the algebraic formula (1.7) stated below (this was partially carried out by Dewar and Murty [23]). The equivalence of (1.1) and (1.7) (in a more general setting) is made explicit by [7,Proposition 7].…”

On class invariants for non-holomorphic modular functions and a question of Bruinier and Ono

“…denote the usual fundamental domain for the action of SL 2 (Z) on H. Our main analytic result is then the following, which is an effective form of a result of Dewar and Murty (see Lemma 5 in [5]).…”

“…Using this,we can directly calculate the Fourier expansion of F at all cusps (correcting a typo in (3.2) of [5]). To this end we choose the following 12 right coset representatives of SL 2 (Z)/Γ 0 (6) (where T := ( 1 1 0 1 )), (3.1)…”

“…By Proposition 3.2, this is equivalent to showing that there is at least one singular modulus P (τ Q ) which is assumed only once as Q ranges over P δ . To see this, we use an effective version of Lemma 5 in [5]. For this, let…”