## Abstract

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

by adjusting the well-known approach of Meinardus.

This is a preview of subscription content, access via your institution.

## Notes

- 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.

## References

- 1.
Andrews, G.E.: The Theory of Partitions. Reprint of the 1976 Original. Cambridge Mathematical Library, p. xvi+255. Cambridge University Press, Cambridge (1998)

- 2.
Balasubramanian, R., Luca, F.: On the number of factorizations of an integer. Integers

**11**, A12 (2011). 5 pp - 3.
Granovsky, B.L., Stark, D.: A Meinardus theorem with multiple singularities. Commun. Math. Phys.

**314**(2), 329–350 (2012) - 4.
Luca, F., Ralaivaosaona, D.: An explicit bound for the number of partitions into roots. J. Number Theory

**169**, 250–264 (2016) - 5.
Meinardus, G.: Asymptotische aussagen über partitionen. Math. Z.

**59**, 388–398 (1954)

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## 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

where

Let us write

Then (A.1) becomes

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

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

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

Hence,

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

## Rights and permissions

## About this article

### Cite this article

Chern, S. Note on square-root partitions into distinct parts.
*Ramanujan J* **54, **449–461 (2021). https://doi.org/10.1007/s11139-019-00191-8

Received:

Accepted:

Published:

Issue Date:

### Keywords

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

### Mathematics Subject Classification

- 11P82
- 05A17