The asymptotic formulas for coefficients and algebraicity of Jacobi forms expressed by infinite product

On the algebraicity about the Hodge numbers of the Hilbert schemes of algebraic surfaces

“…In fact, this is the left-hand side of Theorem 1.1 with x = e 2πiz1 , y = e 2πiz2 , and q = e 2πiτ . Our first theorem is a generalization of our previous result [16,Theorem 1.2]. We say that α is a unit over a commutative ring R with unity, if α and α −1 are integral over R. Theorem 1.2.…”

“…We remark that the two generating functions used in the above two papers [7,20] can be combined mathematically to a general generating series, as was later given by the authors [16] where the general asymptotic formula was obtained. We also refer the reader, to the papers written by Gillman-Gonzalez-Schoenbauer [8] and Gillman-Gonzalez-Ono-Rolen-Schoenbauer [9], respectively, where some specialized cases corresponding to (x, y) = (±1, ±1) (respectively (x, y) = (e…”

“…In our previous work [16], it was shown that for a smooth projective surface S the value e − πiτ 12 χ(S) η(τ…”

Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions

“…They in [10] proved the asymptotics for the cases of fixed m, and in [8] proved the uniform asymptotic formula for a m,k (n) in which holds for all |m| ≤ n/(6kπ 2 ) log n as n tends to infinity. The more related investigations, see for examples, [12,30,27].…”

“…In this paper, we investigate the more precise uniform asymptotic behavior of j m,k (n), a m,k (n) and b m,k (n), which is partly suggested Bringmann and Manschot [10, Section 1.2]. Our main tool is the classical asymptotic analysis which is different above literature [10,8,30,27]. We note that all of those are used the circle method.…”