Disproof of a conjecture by Rademacher on partial fractions
By Michael Drmota and Stefan Gerhold
Abstract
In his book Topics in Analytic Number Theory (1973), Hans Rademacher considered the generating function of integer partitions into at most $N$ parts and conjectured certain limits for the coefficients of its partial fraction decomposition. We carry out an asymptotic analysis that disproves this conjecture, thus confirming recent observations of Sills and Zeilberger (Journal of Difference Equations and Applications19 (2013)), who gave strong numerical evidence against the conjecture.
1. Introduction
In his book Topics in Analytic Number TheoryReference 13, Hans Rademacher gave a partial fraction decomposition of the partition generating function $\prod _{j\geq 1}(1-x^j)^{-1}$. He conjectured that the decomposition of the generating function of partitions into at most $N$ parts (equivalently, into parts from $\{1,\dots ,N\}$),
is consistent with it in the sense that the coefficients $C_{h,k,l}(N)$ converge for $N\to \infty$ to the coefficients of the decomposition of the unrestricted generating function. Despite attracting the attention of several authors Reference 1Reference 3Reference 5Reference 10, the conjecture has been open at least since the early 1960s, when Rademacher mentioned it in a lecture. See Sills and Zeilberger Reference 15 for some further historical remarks. The latter paper presents a recurrence for $C_{0,1,l}(N)$; the values computed by it do not seem to show convergence, but rather oscillating and unbounded behavior. It is well known, though, that there are number-theoretic problems where the true asymptotics are numerically visible only for very large values. See, e.g., Section 2 of Reference 7 for an example. The present note rigorously confirms the main observation from Reference 15, i.e., we disprove Rademacher’s conjecture.Footnote1 To formulate our main result, recall the definition of the dilogarithm function: $\mathrm{Li}_2(w) = \sum _{k\geq 1}w^k/k^2$,$|w|<1$. Define $z_0\approx -1.61 + 7.42i$ as the solution of
1
It is important to note that Cormac O’Sullivan has disproved Rademacher’s conjecture independently from us, as announced in his paper Reference 11, with a different approach. More precisely, he proved that there exist $h,k \le 100$ such that $C_{h,k,1}(N)$ does not converge (personal communication), whereas our method proves directly a conjectural relation from Reference 11, (Conjecture 6.2) – with a slightly worse error term.
(It is easy to show that there is a unique root within, say, distance $1$ of the numerical value given above.) Furthermore, define $\rho =\exp (i a)$, where
Note that the number under the first radical sign is real and positive. The period $p$ of the oscillations is roughly $32$, as observed by Sills and Zeilberger Reference 15. It is independent of $l$, as is the exponential growth order $b^N$. Moreover, Sills and Zeilberger found that the successive peaks seem to grow exponentially with a factor around $8$. The (asymptotically) true factor is $b^p\approx 8.81$. Figures 1 and 2 illustrate the quality of the approximation, which seems to be better for $l=1$ than for $l=2$. Note also that the exponent of $N$ in the error estimate of Equation 3 can certainly be improved.
In principle, it should be possible to extend our approach from $(h,k)=(0,1)$ to general $h,k$. Moreover, a natural conjecture is that the period $p=2\pi /|\arg (1-e^{z_0})|$ of $H_l$ is a transcendental number. While there is some literature on transcendence of polylogarithm values (see, e.g., Reference 8), we are not aware of any result that would imply this.
The rest of the paper is organized as follows. In Section 2, we appeal to the Cauchy integral representation of $C_{0,1,l}(N)$ and find an asymptotic approximation for its integrand by a Mellin transform approach. The new integrand is analysed in Section 3 by the saddle point method. Section 4 completes the proof of Theorem 1 by adding estimates in regions where the asymptotic approximation for the integrand has to be modified or is invalid. In the conclusion, we comment on the error term in Equation 3, and on possible future work.
2. Mellin transform asymptotics
Since the $C_{0,1,l}(N)$ are the Laurent coefficients of $\prod _{j=1}^N (1-x^j)^{-1}$ at $x=1$, we can express them by Cauchy’s formula:
Here, the radius of the integration circle may be any member of $(0,2\pi )$. We wish to replace the integrand $F$ by an asymptotic approximation, derived by Mellin transform asymptotics. We do the analysis for $\Re z<0$, since the factor $e^{-Nz/2}$ in
suggests that the contribution of the left half-circle dominates the integral Equation 5; a rigorous argument is given in Section 4. To take the Mellin transform of $f=\log F$ w.r.t. $N$, we have to interpolate between integral values of $N$. We therefore rewrite the logarithm of the product $\prod _{j=1}^N$ in Equation 4 as follows:
Recall that the polylogarithm is defined for $|w|<1$ and $\nu \in \mathbb{C}$ by $\mathrm{Li}_\nu (w) = \sum _{k\geq 1}w^k/k^\nu$. For the integral evaluation used in the third equality, see Titchmarsh Reference 16, p. 18; it already appears in Riemann’s original memoir Reference 14. By the Mellin inversion formula Reference 6, $g$ can be recovered by
We now move the integration line to the right and collect residues. To estimate the resulting integral (and justify Mellin inversion), we first establish a uniform bound on $\mathrm{Li}_{1-s}(e^z)$ for $|\Im s|$ large. Note that Pickard Reference 12 studied asymptotics of $\mathrm{Li}_\nu (w)$ for $\nu \to 0$ and $\nu \to \infty$, and wrote that “little is known about behavior in the $\nu$-plane except along and near the line $(0,\infty )$.”
We can now find the asymptotics of $g$ (and thus of $f$) by shifting the integration in Equation 8 to the right, where $\Re s =8/7$ turns out to be a suitable choice. The polylogarithm $\mathrm{Li}_{1-s}(e^z)$ is an entire function of $s$. Moreover, $\zeta (-s)$ has a simple pole at $s=-1$, and $\Gamma (-s)$ has simple poles at the non-negative integers. Because of the factor $\zeta (1-s)$, the transform Equation 7 has a double pole at $s=0$, which results in a logarithmic term in the asymptotics of $g$. For fixed$z$, the analysis would be entirely straightforward, but we need some uniformity w.r.t. the integration variable $z$.
Lemma 3 suggests the approximate integral representation
where $h$ from Equation 11 has been replaced by zero, except the term $z/N$, which was retained for better accuracy. Recall that the right half-circle is negligible, as mentioned above and proved in Section 4. Even for small $N$, the fit is very good for $l=1$; see Figure 3.
3. Saddle point asymptotics
We now proceed by a saddle point analysis of the integral Equation 5, using the approximation of the integrand provided by Lemma 3. According to this lemma, the factor $\exp \left( \frac{1}{z} \left( \mathrm{Li}_2(e^z) - \frac{\pi ^2}{6}\right) N \right)$ dominates the integrand in Equation 5. Equating its derivative to zero, we obtain the saddle point $z_0$ defined in Equation 1. The argument of its axis is (see Reference 4)
and $\rho =e^{ia}$ is thus the direction of steepest descent. By symmetry, the conjugate $\bar{z}_0$ is a saddle point, too, and its direction of steepest descent is $\bar{\rho }$. We now deform the integration
circle as follows (see Figure 4): In the right half-plane, we stay with a half-circle, of radius $5$. In the left half-plane, we connect the point $z_1:=5i$ with the point $z_0-\rho$ by a straight line. We then proceed by a line through the saddle point $z_0$, up to a point $z_4(N)$. A vertical line then connects this point to the real axis, to $z_5(N):=-\sqrt {N}$, and so $z_4$ must be
In the lower half-plane, the contour is defined symmetrically. We refer to the line from $z_2$ to $z_3$ to the (upper) central part of the contour, as it gives the dominant contribution to the integral (in the upper half-plane). Note that $-39/112\approx -0.348$ is just a little bit less than $-1/3$. To make the third-order term $N(z-z_0)^3$ in the local expansion of the integrand negligible, we must have $z-z_0 \ll N^{-1/3}$. It is convenient to make the central part as large as possible, though, because this causes faster decrease (as $N\to \infty$) of $F=e^f$ at $z=z_2(N)$ and $z=z_3(N)$, which in turn makes it easier to beat the estimate for $h$ from Lemma 3. (For details, see the tail estimate in Lemma 4 below.)
Part (i) of Lemma 3 provides the local expansion in the central range, where $z=z_0+t\rho$,$-N^{-39/112}\leq t \leq N^{-39/112}$:
$$\begin{multline} f(z,N) = - N \log (1 -e^{z_0}) -\tfrac{1}{2} \alpha N t^2 - \tfrac{1}{2} \log N \\ + (l-\tfrac{1}{2})\log (-z_0) -\tfrac{1}{2} (\log 2\pi +\log (1-e^{z_0})) + O(N^{-5/112}). \cssId{texmlid18}{\tag{18}} \end{multline}$$
(Note that the expansion was simplified by using the defining equation Equation 1 of $z_0$.) The constant
To see that the two small line segments containing the saddle points $z_0$ resp. $\bar{z}_0$ capture the asymptotics of the full integral Equation 5, we have to show that the remaining part of the contour in Figure 4 is negligible. By conjugation, it clearly suffices to consider the upper half-plane. We begin with the part that, additionally, lies in the half-plane $\Re z \leq -N^{-7/8}$. In the next section, we show that the integral over the remaining part tends exponentially to zero.
4. Estimates close to the imaginary axis and in the right half-plane
In the preceding section, we gave an asymptotic evaluation of the integral Equation 5, where the contour was deformed as in Figure 4, and $\Re z < -N^{-7/8}$. We now show that the remaining part of the contour is negligible. Close to the imaginary axis, where $-N^{-7/8} \leq \Re z \leq 0$, we are outside of the validity region of the Mellin transform asymptotics of Lemma 3. We thus estimate the integrand in Equation 5 directly.
Finally, we estimate the integral over the right half-circle in Equation 5, which completes the proof of Theorem 1. By the reflection formula Equation 6, we can recycle part of the analysis from the left half-plane.
5. Conclusion
The error term that we obtained in Equation 3 can be improved a bit by considering more terms of the local expansion Equation 18 of $f$ in the saddle point analysis. Getting the correct order of the error term, i.e., the next term in the asymptotic expansion of $C_{0,1,l}(N)$, needs some work, though. As only the first term on the right hand side of Equation 11 was used to define the saddle point $z_0$, the logarithmic terms in Equation 11 contribute a non-vanishing first order term $O(z-z_0)=O(N^{-39/112})$ to the expansion Equation 18. To improve it, we need to replace $z_0$ by a better approximation of the actual saddle point of the whole integrand $F=e^f$. But then, the tail estimate in Lemma 4 becomes more involved, because not only the width, but also the location of the saddle point segment depends on $N$.
Perhaps more importantly, we comment on possible future work. Recall that Rademacher’s conjecture essentially says that the operations of limit and partial fraction decomposition commute in the present setting. While our result refutes the conjecture, it does not clarify the relation between the partial fraction decompositions of $\prod _{j\geq 1}(1-x^j)^{-1}$ and $\prod _{j=1}^N(1-x^j)^{-1}$. It would be surprising if there was none at all. Maybe there is a summation method that yields convergence. According to numerical evidence, Cesàro summation does not seem to be appropriate.
Acknowledgement
The authors thank the referee for a careful reading of the manuscript and for valuable remarks.
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The authors gratefully acknowledge financial support from the Austrian Science Fund (FWF) under grants P 24880-N25 (second author), resp. F5002 (first author).
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