Note on square-root partitions into distinct parts


Let \(r_D(n)\) be the number of square-root partitions of n into distinct parts. We will give the asymptotic formula of \(r_D(n)\),

$$\begin{aligned} r_D(n)&\sim 2^{-7/6}3^{-1/3}\pi ^{-1/2}\zeta (3)^{1/6}n^{-2/3}\\&\quad \times \exp \Bigg (\frac{3^{4/3}\zeta (3)^{1/3}}{2}n^{2/3}+\frac{\zeta (2)}{2\cdot 3^{1/3}\zeta (3)^{1/3}}n^{1/3}-\frac{\zeta (2)^2}{72\zeta (3)}\Bigg ), \end{aligned}$$

by adjusting the well-known approach of Meinardus.

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  1. 1.

    In fact, if we define \(\mu (\sigma ):=\inf \big \{m\in \mathbb {R}:\zeta (\sigma +it)=O(|t|^m)\big \}\), then

    $$\begin{aligned}\mu (\sigma )={\left\{ \begin{array}{ll} 0 &{} \text {if }\sigma >1,\\ \frac{1}{2}(1-\sigma ) &{} \text {if }0\le \sigma \le 1,\\ \frac{1}{2}-\sigma &{} \text {if }\sigma <0. \end{array}\right. }\end{aligned}$$

    When \(\sigma >1\), it is trivial. When \(\sigma <0\), the result follows from the functional equation of the Riemann zeta function. When \(0\le \sigma \le 1\), the result can be deduced from the theorem of Phragmén–Lindelöf.


  1. 1.

    Andrews, G.E.: The Theory of Partitions. Reprint of the 1976 Original. Cambridge Mathematical Library, p. xvi+255. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  2. 2.

    Balasubramanian, R., Luca, F.: On the number of factorizations of an integer. Integers 11, A12 (2011). 5 pp

    MathSciNet  Article  Google Scholar 

  3. 3.

    Granovsky, B.L., Stark, D.: A Meinardus theorem with multiple singularities. Commun. Math. Phys. 314(2), 329–350 (2012)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Luca, F., Ralaivaosaona, D.: An explicit bound for the number of partitions into roots. J. Number Theory 169, 250–264 (2016)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Meinardus, G.: Asymptotische aussagen über partitionen. Math. Z. 59, 388–398 (1954)

    MathSciNet  Article  Google Scholar 

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Appendix A: Expansion of \(X^{-1}\)

Appendix A: Expansion of \(X^{-1}\)

In this appendix, we give the expansion of \(X^{-1}\). Recall from (2.5) that

$$\begin{aligned} aX^{-3}+bX^{-2}-n=0, \end{aligned}$$


$$\begin{aligned} a=3\zeta (3) \quad \text {and}\quad b=\frac{\zeta (2)}{2}. \end{aligned}$$

Let us write

$$\begin{aligned} \mu =n^{-\frac{1}{3}}\quad \text {and}\quad X^{-1}=a^{-\frac{1}{3}}\mu ^{-1}+\xi . \end{aligned}$$

Then (A.1) becomes

$$\begin{aligned} a\big (a^{-\frac{1}{3}}\mu ^{-1}+\xi \big )^{3}+b\big (a^{-\frac{1}{3}}\mu ^{-1}+\xi \big )^{2}-\mu ^{-3}=0, \end{aligned}$$

so that by multiplying by \(a^{\frac{2}{3}}\mu ^2\) on both sides, one has

$$\begin{aligned} a\xi \cdot \Big (\big (1+a^{\frac{1}{3}}\mu \xi \big )^2+\big (1+a^{\frac{1}{3}}\mu \xi \big )+1\Big )+b\big (1+a^{\frac{1}{3}}\mu \xi \big )^{2}=0. \end{aligned}$$

Now we may treat \(\xi :=\xi (\mu )\) as an implicit function of \(\mu \) defined by (A.2). Note that

$$\begin{aligned} \xi (0)=-\frac{b}{3a}=-\frac{\zeta (2)}{18\zeta (3)}. \end{aligned}$$

The implicit function theorem ensures that we may write \(\xi (\mu )\) as a power series in \(\mu \) in a neighborhood of \(\mu =0\). We compute that

$$\begin{aligned} \xi ^\prime (0)=\frac{b^2}{9 a^{5/3}}=\frac{\zeta (2)^2}{36(3\zeta (3))^{5/3}}. \end{aligned}$$


$$\begin{aligned} \xi (\mu )=-\frac{\zeta (2)}{18\zeta (3)}+\frac{\zeta (2)^2}{36(3\zeta (3))^{5/3}}\mu +O(\mu ^2) \end{aligned}$$

so that as \(n\rightarrow \infty \),

$$\begin{aligned} X^{-1}=\frac{1}{(3\zeta (3))^{1/3}}n^{\frac{1}{3}}-\frac{\zeta (2)}{18\zeta (3)}+\frac{\zeta (2)^2}{36(3\zeta (3))^{5/3}}n^{-\frac{1}{3}}+O(n^{-\frac{2}{3}}). \end{aligned}$$

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Chern, S. Note on square-root partitions into distinct parts. Ramanujan J 54, 449–461 (2021).

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  • Square-root partition
  • Distinct part
  • Asymptotic formula
  • Saddle point method
  • Mellin transform

Mathematics Subject Classification

  • 11P82
  • 05A17