On the asymptotic distinct prime partitions of integers

TL;DR

This paper derives an improved asymptotic formula for the number of ways to write an integer as a sum of distinct primes, including higher-order corrections, which significantly enhances approximation accuracy.

Contribution

It introduces a refined asymptotic expression for $Q(n)$ using saddle-point approximation and higher-order corrections, surpassing previous simple exponential estimates.

Findings

The new asymptotic form closely matches the exact $Q(n)$ for large $n$.

Higher-order corrections improve the approximation accuracy.

The method involves Laplace inversion of the fermionic partition function.

Abstract

We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple leading-order exponential form given so far in the literature.

Equations

$$E=\sum _p{n}_{p}p.$$

$$\sum _p{n}_p=N,n_p=0,1.$$

$$Z^F(\beta )=\prod _p[1+e^{-\beta p}]$$

$$Z^F(\beta )=\sum _{n=0}^{\infty }Q(n)e^{-n\beta },$$

$${\rho }^F(E)=\frac{1}{2\pi i}{\int }_{-i\infty }^{i\infty }d\beta Z^F(\beta )e^{\beta E}.$$

$${\rho }^F(E)=\sum _{n=0}^{\infty }Q(n)\delta (E-n),$$

$$⟨{\rho }^F(E){⟩}_{\Delta E}\sim ⟨Q(n){⟩}_{\Delta n}forE,n\to \infty .$$

$${\rho }^F(E)=\frac{1}{2\pi i}{\int }_{-i\infty }^{+i\infty }d\beta e^{\beta E+\ln Z^F(\beta )}.$$

$$S^F(E,\beta )=\beta E+\ln Z^F(\beta ).$$

$${\frac{\partial S^F(E,\beta )}{\partial \beta }|}_{{\beta }_0}=E+\frac{\partial Z^F(E,{\beta }_0)/\partial \beta }{Z^F(E,{\beta }_0)}=0.$$

$$S^{F(n)}(E,{\beta }_0)={\frac{{\partial }^n{S}^F(E,\beta )}{\partial {\beta }^n}|}_{{\beta }_0}.$$

$${\rho }_{as}^F(E)=\frac{e^{S^F(E,{\beta }_0)}}{\sqrt{2\pi S^{F(2)}(E,{\beta }_0)}}\left[1+\cdots \right],$$

$$\ln Z^F(\beta )=\sum _{p=2}^{\infty }\ln \left(1+e^{-\beta p}\right).$$

$$\ln Z^F(\beta )\sim {\int }_2^{\infty }dx{g}_{av}(x)\ln \left(1+e^{-\beta x}\right),$$

$$\ln Z^F(\beta )\sim \frac{1}{\beta }{\int }_{2\beta }^{\infty }dy\frac{1}{\ln (\frac{y}{\beta })}\ln \left(1+e^{-y}\right)=-\frac{1}{\beta \ln \beta }{\int }_{2\beta }^{\infty }dy\frac{1}{1-\frac{\ln (y)}{\ln (\beta )}}\ln \left(1+e^{-y}\right).$$

$$\ln Z^F(\beta )\sim -\frac{1}{\beta \ln \beta }{\int }_{2\beta }^{\infty }dy\left[1+\sum _{k=1}^{\infty }{\left(\frac{\ln (y)}{\ln (\beta )}\right)}^k\right]\ln \left(1+e^{-y}\right).$$

$$\ln Z_{as}^F(\beta )=-\frac{1}{\beta \ln \beta }{\int }_0^{\infty }dy\left[1+\left(\frac{\ln (y)}{\ln (\beta )}\right)\right]\ln \left(1+e^{-y}\right).$$

$$\ln Z_{as}^F(\beta )=\frac{1}{\beta \ln (\beta )}\left[-\frac{{\pi }^2}{12}+\frac{1}{\ln \beta }\left(\frac{C{\pi }^2}{12}+\sum _k(-1{)}^{k-1}\frac{\ln (k)}{k^2}\right)\right],$$

$$S^F(E,\beta )=\beta E-\frac{F_1}{\beta \ln (\beta )}+\frac{F_2}{\beta (\ln \beta {)}^2}+\cdots$$

$$F_1=\frac{{\pi }^2}{12},F_2=\left[\frac{C{\pi }^2}{12}+\sum _{k=1}^{\infty }(-1{)}^{k-1}\frac{\ln (k)}{k^2}\right]$$

$$F_2=\frac{{\pi }^2}{12}[C-\ln (2)]-\frac{{\zeta }^{\prime }(2)}{2}=0.3734242774\dots$$

$$S^B(E,\beta )=\beta E-\frac{f_1}{\beta \ln (\beta )}+\frac{f_2}{\beta (\ln \beta {)}^2}+\cdots ,$$

$$f_1=\frac{{\pi }^2}{6},f_2=\left[\frac{C{\pi }^2}{6}+\sum _k\frac{\ln (k)}{k^2}\right]$$

$$f_2=\frac{C{\pi }^2}{6}-{\zeta }^{\prime }(2)=\frac{{\pi }^2}{6}[12\ln (A)-\ln (2\pi )]=1.887029965\dots ,$$

$${\rho }_{as}^F(E)=\frac{1}{\sqrt{4E^{3/2}[6\ln (E){]}^{1/2}}}\exp \left\{2\pi \sqrt{\frac{E}{6\ln (E)}}\left[1-\frac{1}{2}\frac{\ln [\ln (E)]}{\ln (E)}+b^F\frac{1}{\ln (E)}\right]\right\}$$

$$b^F=\left[\frac{F_2}{F_1}+\ln (\pi /\sqrt{6})\right]=0.7028796287\dots$$

$$Q_{as}(n)=\frac{1}{\sqrt{4n^{3/2}[6\ln (n){]}^{1/2}}}\exp \left\{2\pi \sqrt{\frac{n}{6\ln (n)}}\left[1-\frac{1}{2}\frac{\ln [\ln (n)]}{\ln (n)}+b^F\frac{1}{\ln (n)}\right]\right\}.$$

$$P_{as}(n)=\frac{1}{\sqrt{4n^{3/2}[3\ln (n){]}^{1/2}}}\exp \left\{2\pi \sqrt{\frac{n}{3\ln (n)}}\left[1-\frac{1}{2}\frac{\ln [\ln (n)]}{\ln (n)}+b^B\frac{1}{\ln (n)}\right]\right\},$$

$$b^B=\left[\frac{f_2}{f_1}+\ln (\pi /\sqrt{3})\right]=0.7426003995\dots$$

$$Q_0(n)=\exp \left\{2\pi \sqrt{\frac{n}{6\ln (n)}}\right\},$$

Full text

I Introduction

The asymptotic form $Q_{as}(n)$ of the distinct prime partitions $Q(n)$, called the sequence A000586 in oeis , has in the literature so far only been given in its leading exponential form roth , published as long ago as in 1954. We believe that it is time to improve the asymptotics by corrections to the leading exponential form which itself is rather poor oeis . Although this might appear as a straight-forward task, it requires quite some cumbersome algebraic efforts. For the unrestricted prime partions $P(n)$, called the sequence A000607 in oeis , the next-to-leading order correction has been derived and published in 2008 in this journal vaughan . The numerical coefficient of the next-to-leading order term was corrected recently in BBBM . For our present investigations we will employ here the same method as in BBBM . Although this method algebraically is very close to that used in vaughan , we present it as viewed from the standpoint of statistical mechanics. The fact that the two partitions mentioned above can be connected to many-body systems of particles obeying opposite statistics may, in our opinion, serve as a bridge between mathematical physics and pure number theory.

It is well established by now muoi that the techniques of statistical mechanics can be applied to obtain any type of partition of a positive integer $n$. The partition function of a gas in statistical mechanics contains information on the distribution of the total energy among the constituents and hence plays the same role as the generating function of the corresponding partitions in number theory. This relation was used in muoi , where the number of partitions $P(n)$ known from number theory is obtained from the quantum density of states $\rho(E)$ given by the inverse Laplace transform of the partition function. Taking the Laplace transform approximately using the saddle-point method then yields the asymptotic forms $P_{as}(n)$.

This method was applied in BBBM to the unrestricted partitions $P(n)$ of integers into primes, i.e., the sequence A000607. For a system whose single-particle levels are defined by the primes $p$ as an ordered set, the total energy is given by a sum of primes, and the corresponding density of states is related to the number of unrestricted prime partitions $P(n)$, assuming that the particles behave like bosons. The asymptotic form $P_{as}(n)$ obtained in BBBM was found to approximate the exact $P(n)$ for large $n$ much better than the asymptotic expressions given earlier in the literature yang ; vaughan . The same method was also applied more recently to distinct square partitions in MBBB .

In the present paper we study $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, i.e., the sequence A000586. As only distinct primes are allowed in $Q(n)$, this corresponds to a system of fermionic particles, obeying the Pauli exclusion principle, still with the primes $p$ as single-particle levels. We again use the saddle-point method for Laplace inverting their partition function to derive algebraically the asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. In numerical computations up to $n=10^{5}$ we find that, like in BBBM , our result for $Q_{as}(n)$ approaches the exact $Q(n)$ far better than the simple leading-order exponential form given so far in the literature roth .

The plan of our paper is as follows. In Section II.1, we establish the relation of $Q(n)$ to the partition function and the density of states $\rho^{F}(E)$. In Sec. II.2 we derive its asymptotic form using the saddle-point method, and in Sec. II.3 we give the explicit solution of the saddle-point equation leading to our final result for $Q_{as}(n)$. In Sec. III our asymptotic result is compared numerically with the exact function $Q(n)$ for the distinct prime partitions. We conclude the paper with a short summary in Sec. IV.

II Partitions into primes

II.1 Fermionic partition function and its relation to $Q(n)$

Consider a large number $N$ of fermions whose single-particle spectrum is given by the primes $p$. The total energy $E$ of the system is given by

[TABLE]

(We use throughout dimensionless variables and take the particle mass $m$, the Planck constant $\hbar$ and the Boltzmann constant $k$ to be unity: $m=\hbar=k=1$.) Here and in the following, the sums $\sum_{p}$ run over all primes $p$, and $n_{p}$ are the fermionic occupancies of the levels which must be zero or one, such that

[TABLE]

The number of possible energy partitions $E$ with the restriction (2) shall be denoted by $Q_{N}(E)$, where the subscript $N$ keeps track of the total number of particles. Although $E$ is necessarily integer, we treat it as a continuous variable like in statistical mechanics. $Q_{N}(n)$ is the number of $N$-restricted fermionic partitions of $n$, i.e., the number of ways to write $n$ as a sum of $N$ distinct primes. In the limit $N\to\infty$, $Q_{N}(n)$ will tend towards the number of unrestricted but distinct prime partitions $Q(n)$ under consideration here.

For the purpose of this paper, we are only interested in the limit $N\to\infty$ of the fermionic partitions which then become unrestricted as stated above. The quantum-statistical partition function $Z^{F}(\beta)$ is in this limit given by

[TABLE]

where $\beta=1/kT$ is the inverse temperature and the product runs over all primes $p$. Taylor expanding the expontential in (3) and reordering the terms yields the alternative form of the partition function

[TABLE]

which in number theory is known as the generating function of the $Q(n)$. In the On-line Encyclopedia of Integer Sequences (OEIS) oeis , the sequence of numbers $Q(n)$ is called the sequence A000586. Its first ten members are $Q(n)$ = 1, 0, 1, 1, 0, 2, 0, 2, 1, 1 for $n=0,...,9$, where $Q(0)=1$ by definition. Note also that the $Q(n)$ are a subset of the (bosonic) prime partitions $P(n)$, called the sequence A000607 in oeis .

From the partition function, we obtain the many-body density of states $\rho^{F}(E)$ by an inverse Laplace transform:

[TABLE]

Hereby $\beta$ is taken as a complex variable and the integration above runs along the imaginary axis of the complex $\beta$ plane. Later the Laplace inversion shall be taken in the saddle-point approximation.

It is important now to realize that $Q(n)$ is related to the density of states $\rho^{F}(E)$ in the following way. Taking directly the exact inverse Laplace transform of (4), we find

[TABLE]

where $\delta(E-n)$ is the Dirac delta function peaked at $E=n$. We see thus that $\rho^{F}(E)$ can also be understood as the density of distinct prime partitions. Like it was argued in MBBB for the distinct square partitions, averaging $\rho^{F}(E)$ over a sufficiently large energy interval $\Delta E$ is asymptotically the same as averaging $Q(n)$ over a sufficiently large interval $\Delta n$:

[TABLE]

Therefore determing the asymptotic average part $\rho_{as}^{F}(E)$ of the density of states valid in the limit $E\to\infty$, which can be obtained by the saddle-point approximation to its inverse Laplace transform (5), and equating $E=n$ will give the average asymptotic form $Q_{as}(n)$ of the distinct prime partitions.

II.2 Asymptotic partition function from saddle-point approximation

We first rewrite the inverse Laplace transform (5) by taking the natural log of $Z^{F}$ into the exponent:

[TABLE]

We now evaluate this integral using the saddle-point method (also called the method of steepest descent). We define the exponent above as the canonical entropy function

[TABLE]

Applying the saddle-point method to (8) requires to find a stationary point $\beta_{0}$ of the function $S^{F}(E,\beta)$ by solving the saddle-point equation

[TABLE]

If this equation has a solution $\beta_{0}$, which will be a function $\beta_{0}(E)$, one evaluates the successive partial derivatives of $S^{F}(E,\beta)$ at $\beta_{0}$:

[TABLE]

The approximate result of the inverse Laplace transform then is given by

[TABLE]

where the dots indicate the so-called cumulants involving higher derivatives of the entropy, which become more important for large $\beta$ (see, e.g., Ref. jelovic ). Since we are interested here in the limit $\beta\to 0$ relevant for the asymptotics of large $E$, we can neglect these cumulants.

Next, we take the natural log of the partition function given in Eq.(3)

[TABLE]

and approximate it by the integral

[TABLE]

where $g_{av}(x)=\frac{1}{\ln(x)}$ is the approximate density of primes, using the prime number theorem. If the density $g_{av}(x)$ were exact, then the integral would give the exact result (13).

The evaluation the integral in the limit $\beta\to 0$ follows closely the method outlined in BBBM . Denoting $y=\beta x$, the integral becomes

[TABLE]

In the limit $\beta\to 0$ we may write this integral as an asymptotic series

[TABLE]

This is now a series in the expansion parameter $1/\ln(\beta)$ since each term is divided by the power $(\ln\beta)^{k}$. As we shall see later, in the leading saddle-point approximation $\ln(\beta_{0})\approx\ln(E)$ and hence this is an asymptotic series in $1/\ln(E)$ as well. In the asymptotic limit we take the lower limit of the integral to be zero.

For the present analysis, we retain the leading term and the first correction, like for the bosonic prime partitions in BBBM , and define

[TABLE]

The integrals may again be evaluated analytically and we obtain

[TABLE]

where $C=0.5772156649\!\dots$ is the Euler constant.

II.3 Solution of saddle-point equation and $Q_{as}(n)$

In order to find the saddle point $\beta_{0}$ from Eq. (10), we start from the entropy $S^{F}(\beta)$ in the asymptotic limit. Using Eqs. (9) and (18) we get up to order $1/(\ln\beta)^{2}$

[TABLE]

where

[TABLE]

The infinite sum in $F_{2}$ may be expressed in a closed form in terms of a derivative of the Riemann zeta function, leading to

[TABLE]

Eq. (19) is identical in form with that of the bosonic case given in BBBM :

[TABLE]

where

[TABLE]

The constant $f_{2}$ may also be expressed in a closed form by

[TABLE]

where $A=1.282427129100$… is the Glaisher-Kinkelin constant (see A074962 in oeis ).

The only difference in going from bosonic to the fermionic case is that the coefficients $f_{1}$ and $f_{2}$ of BBBM are replaced here by the $F_{1}$ and $F_{2}$, respectively. Therefore we obtain our result simply by replacing the coefficients $f_{i}$ in the bosonic case by the $F_{i}$ in the present fermionic case and following the steps outlined in BBBM .

Thus we can directly give the result for the fermionic case as

[TABLE]

with the constant

[TABLE]

The asymptotic $Q_{as}(n)$ is then obtained replacing $E$ by $n$ above, so that:

[TABLE]

This is the main result of the present paper. The corresponding result for the bosonic partitions in BBBM was

[TABLE]

with the constant

[TABLE]

Note that the leading exponential terms and the denominators of the pre-exponential terms in (27) and (28) differ by a factor $1/\!\sqrt{2}$. Note that since the $Q(n)$ are a subset of the $P(n)$, their values must be smaller, which asymptotically is brought about by the extra factor $1/\!\sqrt{2}$ in the leading exponential term. The first correction term in the exponent, namely $-\frac{1}{2}\ln[\ln(n)]/\!\ln(n)$, is identical in both cases. As far as we know, the above result (27) for the distinct prime partitions has not been given in the literature so far.

In the next section, we compare numerically our asymptotic result (27) with the exact values $Q(n)$ of the distinct prime partitions.

III Numerical test of $Q_{as}(n)$

In this section we test our asymptotic result (27) numerically. We have generated the exact $Q(n)$ up to $n=100\,000$. In Figs. 1 and 2 we show their values by the dots (red) on a logarithmic scale in two regions of $n$. The dashed line (green) shows the leading-order exponential expression

[TABLE]

which is the only asymptotic result that has been given so far in the literature roth , and the solid (blue) line gives our full asymptotic result (27).

A large discrepancy between $Q_{0}(n)$ and $Q(n)$ is noticed for all $n$. Our full asymptotic result $Q_{as}(n)$ (27) approaches the exact $Q(n)$ much better (except in the academic limit $n\to 0$ where it diverges due to the pre-exponential factor). In Fig. 2 for the values $n\geq 20\,000$, the two curves can hardly be distinguished.

We have thus achieved a considerable improvement over the simple exponential form (30). A closer look reveals that the curve for $Q_{as}(n)$, which for smaller $n$ overestimates the exact $Q(n)$, crosses the curve of the latter around $n\sim 50\,000$. A similar result was found in BBBM for the bosonic prime partitions, where $P_{as}(n)$ crosses $P(n)$ much earlier and then tends to approach it asymptotically from below for $n\to\infty$.

In order to focus on this asymptotic behavior, we show in Fig. 3 the difference of the natural logs relative to the lowest-order term, i.e., the quantity $[\ln Q_{as}(n)-\ln Q(n)]/\ln Q_{0}(n)$, plotted versus $1/n$ in a region of the largest $n$ available. The solid (red) curve gives the result obtained with our full asymptotic form (27). For comparison we show in this figure by the dotted (blue) curve also the corresponding quantity obtained in Ref. BBBM from the unrestricted (bosonic) prime partitions $P(n)$ and their respective asymptotic forms. The overall behaviour of the two curves is similar. For the results in BBBM we had larger values of $n$ available. There we noticed a tendency for the difference to approach zero from below for $1/n\to 0$ (i.e. $n\to\infty$), as can be recognized from the blue curve in Fig. 3.

Note added after completion of our work:

After the publication of our results on the arXiv server in 2019 MBBarx , V. Kotesovec has performed numerical studies of our $\ln Q_{as}(n)$ and the exact $\ln Q(n)$, computing these quantities up to $n_{max}=10^{8}$. In order to achieve this, he programmed in the assembler a special floating point arithmetic in which both the mantissa and the exponent of these quantities have 8 bytes. By this procedure it was possible to generate 10 million terms in 9 hours, and the calculation of $n_{max}=10^{8}$ terms took 31 days.

V. Kotesovec has kindly sent us his results which confirm our findings (red line) up to $n=10^{5}$ and furthermore show that, indeed, the difference $[\ln Q_{as}(n)-\ln Q(n)]/\!\ln Q_{0}(n)$ tends towards zero from the same side as that of the bosonic (unrestricted) prime partitions (blue curve). Similarly to the situation for the latter, the asymptotic ratio $\ln Q(n)/\!\ln Q_{as}(n)$ first exceeds the value 1 but then reaches a maximum, occurring here at $n=14\,474\,250$, has an inflection point at n\,\hbox{\kern 1.00006pt\lower 2.58334pt\hbox{\sim} \kern-11.19997pt\raise 2.58334pt\hbox{>} }33\,272\;000, and gradually decreases back towards 1. One or two million terms are far from enough for this finding; it is necessary to have at least 40 million terms. A graph of Kotesovec’s result for $\ln Q(n)/\!\ln Q_{as}(n)$ is posted at OEIS kotes .

IV Summary

In summary, we have shown how an improved asymptotic expression for the function $Q(n)$, which counts the number of distinct prime partitions of an integer $n$, can be obtained from asymptotic expansions of the partition function $Z^{F}(\beta)$ in (4) and the corresponding density of states $\rho^{F}(E)$ in (5). $Z^{F}(\beta)$ can be understood as the quantum-statistical partition function of a system of $N$ fermions, whose single-particle energy spectrum is given by the primes $p$, in the limit $N\to\infty$. It is identical to the generating function of the $Q(n)$ known in number theory. The density of states $\rho^{F}(E)$ is identical to the the density of distinct prime partitions given in Eq. (6). Exploiting the connection between $\rho^{F}(E)$ and $Q(n)$ using the saddle-point approximation for the inverse Laplace transform (5), we have obtained the asymptotic form $Q_{as}(n)$ in Eq. (27) and shown it numerically to approach the exact $Q(n)$ in the limit $n\to\infty$ far better than the hitherto known expression $Q_{0}(n)$ given in (30).

We have used the same method as in Ref. BBBM where the non-distinct prime partitions $P(n)$ were studied, and have found similar results as there. The asymptotic $Q_{as}(n)$ overestimates the exact $Q(n)$ for smaller $n$ but overshoots it for n\,\hbox{\kern 1.00006pt\lower 2.58334pt\hbox{\sim} \kern-11.19997pt\raise 2.58334pt\hbox{>} }50,000. Like in BBBM , the limit $Q_{as}(n)\to Q(n)$ for $n\to\infty$ cannot be demonstrated rigorously. However, forcing the calculation of our $Q_{as}(n)$ and of $Q(n)$ up to $n=10^{8}$, V. Kotesovec has shown numerically that $Q_{as}(n)$, indeed, approaches $Q(n)$ monotonously for n\,\hbox{\kern 1.00006pt\lower 2.58334pt\hbox{\sim} \kern-11.19997pt\raise 2.58334pt\hbox{>} }4\times 10^{7} kotes . He assumed that the difference can be approximated by a term $c_{2}/\ln^{2}(n)$ in the square brackets of the asymptotic expansion (27) and showed that the results are very sensitive to the value of $c_{2}$. We join his suggestion that the systematic evaluation of the algebraic value of $c_{2}$, or of other correction terms in (27), could be a topic of interesting future research for the next generation of patient researchers.

But already now we can state that already with our present result (27), we have obtained an excellent asymptotic approximation for the distinct prime partitions which is far superior to the hitherto known result roth .

M.V.N.M. and M.B. acknowledge stimulating earlier correspondence with V. Kotesovec and, in particular, the communication of his most recent numerical results. R.K.B. is grateful to the IMSc, Chennai, for its hospitality during the final stages of our collaboration.