Asymptotics and inequalities for the distinct partition function

Abstract In this paper, we give explicit error bounds for the asymptotic expansion of the shifted distinct partition function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {q(n+s)} for any nonnegative integer s . Then based on this refined asymptotic formula, we give the exact thresholds of n for the inequalities derived from the invariants of the quartic binary form, the double Turán inequalities, the Laguerre inequalities and their corresponding companion versions.