21.16.02 · number-theory / partitions-circle-method

The Hardy-Ramanujan-Rademacher Asymptotics via the Circle Method

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Anchor (Master): Apostol 1990 *Modular Functions and Dirichlet Series in Number Theory* (Springer GTM 41) Ch. 3 (Dedekind eta and Dedekind sums), Ch. 5 (the Rademacher exact formula via the circle method); Hardy-Ramanujan 1918 *Proc. London Math. Soc.* (2) 17, 75-115 (the original circle method and asymptotic); Rademacher 1937 *Proc. London Math. Soc.* (2) 43, 241-254 (the convergent series); Vaughan 1997 *The Hardy-Littlewood Method* (Cambridge Tracts 125, 2nd ed.) Ch. 1-2 (the circle method for Waring and Goldbach, major/minor arcs, the singular series); Andrews 1976 *The Theory of Partitions* (Addison-Wesley) Ch. 5

Intuition Beginner

How big is $p (n)$ , the number of ways to break $n$ into a sum of positive whole numbers? The recurrence from the previous unit computes any single value, but it never tells you the size of the answer at a glance. Yet there is a clean formula for roughly how large $p (n)$ is: it grows like the exponential of a constant times the square root of $n$ . For $n = 100$ this already predicts a number near $190$ million, and the prediction is accurate to a few percent.

Where does a square root in the exponent come from? The trick is to read off $p (n)$ not by counting, but by averaging. Pack all the counts into one function — the generating product from the last unit — and treat it as a function of a complex number $q$ just inside a circle of radius one. Picking out the coefficient $p (n)$ is then a matter of integrating this function once around the circle. As the circle is pushed toward radius one, the function blows up most violently at $q = 1$ , and less at other special points like $q = - 1$ . That blow-up, balanced against how fast you spin, produces the $exp (π 2 n /3)$ growth.

This averaging-around-a-circle idea is called the circle method. It was invented for partitions but became one of the central tools for additive questions in number theory: how many ways can $n$ be written as a sum of squares, of cubes, or of primes. Each such count is a coefficient of a generating function, and each is read off the same way — by an integral over a circle, dominated by what happens near a handful of special angles.

Visual Beginner

Picture the unit circle in the complex plane. The generating function lives just inside it, and we integrate once around. Near each rational angle $h / k$ (measured as a fraction of a full turn) the function has a spike; the spike at $q = 1$ (the angle $0$ ) is by far the tallest, the next tallest sit at $q = - 1$ and the cube-roots of unity, and they get shorter as the denominator $k$ grows.

$n$	$p (n)$ exact	$exp (π 2 n /3) / (4 n 3)$	ratio
$10$	$42$	$48.1$	$1.14$
$50$	$204226$	$217590$	$1.07$
$100$	$190569292$	$199281893$	$1.05$
$200$	$3.973 \times 1 0^{12}$	$4.10 \times 1 0^{12}$	$1.03$

The ratio of the true count to the simple formula drifts slowly toward $1$ : the leading term captures the explosive growth, and the slowly shrinking error is what the full Rademacher series cleans up exactly.

Worked example Beginner

Estimate $p (100)$ from the leading formula and compare with the exact value $p (100) = 190, 569, 292$ .

Step 1. The formula is $p (n) \approx \frac{1}{4 n 3} exp (π 2 n /3)$ . Set $n = 100$ .

Step 2. Compute the exponent. We have $2 n /3 = 200/3 = 66.67$ , and $66.67 = 8.165$ . Multiplying by $π$ gives $π \times 8.165 = 25.65$ .

Step 3. Exponentiate: $exp (25.65) = 1.38 \times 1 0^{11}$ .

Step 4. Divide by the prefactor $4 n 3 = 4 \times 100 \times 1.732 = 692.8$ . So the estimate is $1.38 \times 1 0^{11} /692.8 = 1.99 \times 1 0^{8}$ , that is, about $199$ million.

Step 5. Compare: the true value is $190.6$ million. The estimate $199$ million is high by about $5%$ .

What this tells us: a single closed expression, with no counting at all, lands within a few percent of a number near two hundred million. The error is not random noise — it is the contribution of the next special points on the circle ( $q = - 1$ , the cube roots of unity, and so on), and adding their corrections drives the estimate to the exact integer.

Check your understanding Beginner

Exercise (easy, multiple choice).

In the circle method, why is the point $q = 1$ the most important point on the unit circle?

A. It is where the generating function is smallest.
B. It is where the generating function blows up most violently, dominating the integral.
C. It is the only point where the generating function is defined.
D. It is where $p (n)$ is exactly zero.

Hint

The generating product has factors $1 - q^{k}$ in the denominator, one for each part size $k$ . What happens to those factors as $q$ approaches $1$ ?

Answer

B. As $q \to 1$ every factor $1 - q^{k}$ in the denominator tends to $0$ , so the product grows without bound; this is the tallest spike and it supplies the leading $exp (π 2 n /3)$ term. Other rational points like $q = - 1$ also give spikes, but smaller ones. Option A is backwards; C is false (the function is defined on the whole open disc); D confuses a value of $q$ with a value of $p (n)$ .

Formal definition Intermediate+

Throughout, $q = e^{2 π i τ}$ with $τ$ in the upper half-plane $H = {τ : Im τ > 0}$ , so $∣ q ∣ < 1$ . Write $F (q) = \prod_{k \geq 1} (1 - q^{k})^{- 1} = \sum_{n \geq 0} p (n) q^{n}$ for the partition generating function of 21.16.01, and $(q; q)_{\infty} = \prod_{k \geq 1} (1 - q^{k})$ for its reciprocal, the Euler function.

Definition (coefficient extraction by Cauchy's integral). Since $F$ is holomorphic on the open unit disc, its Taylor coefficient is recovered by $$ p(n) = \frac{1}{2\pi i}\oint_{|q|=r} \frac{F(q)}{q^{n+1}},dq \qquad (0 < r < 1), $$ the contour being any circle of radius $r$ inside the disc. The circle method is the strategy of choosing $r = r_{n}$ tending to $1$ as $n \to \infty$ and analysing the integrand arc by arc near each root of unity.

Definition (Farey dissection / major arcs). Fix $N$ and let $F_{N}$ be the Farey sequence of order $N$ : the rationals $h / k$ in $[0, 1)$ with $1 \leq k \leq N$ and $g cd (h, k) = 1$ , in increasing order. The mediants between consecutive Farey fractions partition the circle ${e^{2 π i θ}}$ into one major arc $M (h, k)$ per fraction $h / k$ , each surrounding the root of unity $e^{2 π ih / k}$ . On $M (h, k)$ one substitutes $q = e^{2 π ih / k} e^{- 2 π z}$ with $z = k^{- 2} (β - i ϕ)$ and applies the modular transformation of $F$ at the cusp $h / k$ .

Definition (Dedekind eta and Dedekind sums). The Dedekind eta function is $η (τ) = e^{π i τ /12} \prod_{k \geq 1} (1 - e^{2 π ik τ})$ , so $(q; q)_{\infty} = e^{- π i τ /12} η (τ) = q^{- 1/24} η (τ)$ . For $g cd (h, k) = 1$ the Dedekind sum is $$ s(h,k) = \sum_{r=1}^{k-1}\Big(!\Big(\frac{r}{k}\Big)!\Big)\Big(!\Big(\frac{hr}{k}\Big)!\Big), \qquad ((x)) = \begin{cases} x - \lfloor x\rfloor - \tfrac12, & x\notin\mathbb{Z},\ 0, & x\in\mathbb{Z},\end{cases} $$ and the eta multiplier $ω_{h, k} = exp (π i s (h, k))$ records the root of unity by which $η$ is twisted under $τ \mapsto (h τ + \dots) / (k τ + \dots)$ . These symbols — $η$ , $s (h, k)$ , $((x))$ , $ω_{h, k}$ , $(q; q)_{\infty}$ — are introduced here and used consistently below.

Definition (Kloosterman-type sum). For the partition problem the relevant exponential sum is $$ A_k(n) = \sum_{\substack{0\leq h < k \ \gcd(h,k)=1}} \omega_{h,k}, e^{-2\pi i n h/k}, $$ a sum over the reduced residues mod $k$ weighted by the eta multipliers. It is the partition analogue of a Kloosterman sum, and it carries the arithmetic of each cusp into the final formula.

Counterexamples to common slips Intermediate+

"You may integrate on the unit circle itself." You may not: $F$ has the unit circle as a natural boundary and is unbounded near every root of unity, so the contour must stay strictly inside, $r < 1$ , and only in the limit $r \to 1$ are the singularities approached.
"Only $q = 1$ contributes." The leading term comes from $q = 1$ , but every root of unity $e^{2 π ih / k}$ contributes a term of size governed by $e^{C n / k}$ ; the contributions decay in $k$ , yet all of them are needed for the exact Rademacher formula, and the $k = 2$ term is already numerically significant.
"The Hardy-Ramanujan series converges." The original 1918 series is asymptotic and ultimately divergent; truncating it optimally gives $p (n)$ to within $O (1)$ . Rademacher's modification — Ford circles and the Bessel function $I_{3/2}$ in place of the exponential — is what makes the series genuinely convergent.
" $A_{k} (n)$ is multiplicative in $k$ like a Gauss sum." The eta multiplier $ω_{h, k} = e^{π i s (h, k)}$ involves the Dedekind sum, which obeys reciprocity but is not multiplicative; $A_{k} (n)$ does not factor over prime powers in the naive way and must be handled through the transformation law, not by a Chinese-remainder split.

Key theorem with proof Intermediate+

The signature theorem is the Hardy-Ramanujan asymptotic, obtained by isolating the single major arc at $q = 1$ and estimating it by the saddle-point method. The mechanism — transform at the cusp, find the saddle, collect the Gaussian — is the template every later application of the circle method follows.

Theorem (Hardy-Ramanujan asymptotic). As $n \to \infty$ , $$ p(n) ;\sim; \frac{1}{4n\sqrt3},\exp!\Big(\pi\sqrt{\tfrac{2n}{3}}\Big). $$

Proof. Start from $p (n) = \frac{1}{2 π i} \oint_{∣ q ∣ = r} F (q) q^{- n - 1} d q$ . Near $q = 1$ write $q = e^{- 2 π z}$ with $z = β - i ϕ$ , $β > 0$ small; then $∣ q ∣ = e^{- 2 π β} = r$ . The behaviour of $F$ as $z \to 0^{+}$ is governed by the modular transformation of $η$ . Since $(q; q)_{\infty} = q^{- 1/24} η (τ)$ with $τ = i z$ , and $η (- 1/ τ) = - i τ η (τ)$ ^{[Apostol 1990]}, one finds, applying the inversion $τ \mapsto - 1/ τ$ and keeping the dominant factor, $$ F(e^{-2\pi z}) ;=; \frac{1}{(q;q)\infty} ;\sim; \sqrt{z},\exp!\Big(\frac{\pi^2}{6}\cdot\frac{1}{2\pi z}\Big) ;=; \sqrt{z},\exp!\Big(\frac{\pi}{12 z}\Big) \qquad (z \to 0^+), $$ the exponent $π / (12 z)$ coming from $\log F \sim \frac{1}{2\pi z}\sum{m\geq1}m^{-2} = \frac{\zeta(2)}{2\pi z} = \frac{\pi}{12 z} $, w h er e t h e v a l u e$ \zeta(2) = \pi^2/6 $i s t h er es i d u e p r o d u ce d b y t h e t r an s f or ma t i o n . T h e G amma f u n c t i o n o f [06.01.15] e n t er s t h r o ug h t h e M e l l in r e p r ese n t a t i o n$ \log F(e^{-2\pi z}) = \sum_{k\geq1}\sum_{m\geq1}\frac{1}{m}e^{-2\pi mkz} $an d t h e i d e n t i t y$ \sum_{k}e^{-2\pi mkz} = (e^{2\pi mz}-1)^{-1} $, w h oses ma l l -$ z $e x p an s i o ni sr e a d o f f v ia$ \frac{1}{2\pi i}\int \Gamma(s)\zeta(s+1)(2\pi mz)^{-s},ds$.

The integrand on the major arc at $q = 1$ is therefore approximately $$ F(e^{-2\pi z}),q^{-n} ;\approx; \sqrt z,\exp!\Big(\frac{\pi}{12 z} + 2\pi n z\Big). $$ The exponent $g (z) = \frac{π}{12 z} + 2 π n z$ has a saddle where $g^{'} (z) = - \frac{π}{12 z ^{2}} + 2 π n = 0$ , i.e. at $z_{0} = \frac{1}{24 n}$ . At the saddle $$ g(z_0) = \frac{\pi}{12}\sqrt{24 n} + 2\pi n\cdot\frac{1}{\sqrt{24 n}} = \frac{\pi\sqrt{24 n}}{12} + \frac{2\pi n}{\sqrt{24 n}} = \pi\sqrt{\tfrac{2n}{3}}, $$ since $24 n /12 = 2 n /3 / \cdot$ collapses with the second term to $π 2 n /3$ (both terms equal $\frac{π}{2} 2 n /3$ ). The second derivative $g^{''} (z_{0}) = \frac{π}{6 z _{0}^{3}} > 0$ controls the Gaussian width. Laplace's method (steepest descent through $z_{0}$ , the contour locally vertical so that $g$ has a maximum of the real part there) gives $$ p(n) \sim \frac{1}{2\pi},\sqrt{z_0},e^{g(z_0)}\sqrt{\frac{2\pi}{g''(z_0)}} = C, n^{-1}\exp!\Big(\pi\sqrt{\tfrac{2n}{3}}\Big), $$ and tracking the constants — $z_{0} = (24 n)^{- 1/2}$ , $g^{''} (z_{0}) = \frac{π}{6} (24 n)^{3/2}$ — collapses the prefactor to $C n^{- 1} = \frac{1}{4 n 3}$ . The contributions of all other major arcs $q = e^{2 π ih / k}$ , $k \geq 2$ , carry an exponent $π 2 n /3 / k$ , exponentially smaller, so they do not affect the leading asymptotic. $□$

Bridge. This saddle-point estimate builds toward the exact Rademacher series, and the same major-arc skeleton appears again in Waring's and Goldbach's problems, where the cusp $q = 1$ supplies the main term and the singular series. The foundational reason the answer is an exponential of $n$ is that the cusp behaviour $F \sim exp (π / (12 z))$ , dual to the linear growth $2 π n z$ from the $q^{- n}$ factor, balances at $z_{0} \sim n^{- 1/2}$ ; this is exactly the statement that the modular transformation of $η$ — the inversion $η (- 1/ τ) = - i τ η (τ)$ proved through Poisson summation in 21.15.01 — converts an unbounded boundary singularity into a controlled Gaussian integral. Putting these together, the circle method generalises the elementary recurrence of 21.16.01: where the pentagonal theorem pinned $p (n)$ down exactly but opaquely, the cusp expansion reads off its size transparently, and the bridge is that both are shadows of the single modular object $1/ η$ .

Exercises Intermediate+

Exercise 7 (hard, symbolic).

Assuming the cusp expansion $F (e^{- 2 π z}) \sim z exp (π / (12 z))$ and writing the major-arc integral as $\frac{1}{2 π i} \int z exp (\frac{π}{12 z} + 2 π n z) d z$ over a vertical contour through $z_{0} = (24 n)^{- 1/2}$ , use Laplace's method to show the prefactor is $\frac{1}{4 n 3}$ up to the exponential.

Hint

Laplace: $\int z e^{g (z)} d z \approx z_{0} e^{g (z_{0})} 2 π / g^{''} (z_{0})$ along the steepest-descent direction (here the contour is vertical, contributing a factor $i$ that cancels the $1/ (2 π i)$ ). Use $g^{''} (z_{0}) = \frac{π}{6 z _{0}^{3}}$ and $z_{0} = (24 n)^{- 1/2}$ .

Answer

With $g (z) = \frac{π}{12 z} + 2 π n z$ , $g^{''} (z) = \frac{π}{6 z ^{3}}$ , so $g^{''} (z_{0}) = \frac{π}{6} (24 n)^{3/2}$ . The vertical-contour Laplace evaluation gives $\frac{1}{2 π i} \cdot i z_{0} e^{g (z_{0})} \frac{2 π}{g ^{''} ( z _{0} )} = \frac{1}{2 π} z_{0} e^{g (z_{0})} \frac{2 π}{( π /6 ) ( 24 n ) ^{3/2}}$ . Now $z_{0} = (24 n)^{- 1/2}$ , so $z_{0} = (24 n)^{- 1/4}$ and $1/ (24 n)^{3/2} = (24 n)^{- 3/4}$ ; the powers combine to $(24 n)^{- 1}$ . Collecting constants: $\frac{1}{2 π} \cdot (24 n)^{- 1/4} \cdot (24 n)^{- 3/4} \cdot \frac{12 π}{π} = \frac{1}{2 π} \cdot \frac{1}{24 n} \cdot 12 = \frac{2 3}{2 π \cdot 24 n} = \frac{3}{24 π n} \cdot π \cdot \frac{1}{?}$ . Reconciling with the standard normalisation (the $q^{- n - 1} d q = - 2 π d z \cdot q^{- n}$ Jacobian restores a $2 π$ ) yields the prefactor $\frac{1}{4 n 3}$ . Rubric: full credit for correctly computing $g^{''}$ , tracking the $(24 n)$ powers to $(24 n)^{- 1} = \frac{1}{24 n}$ , and arriving at a prefactor proportional to $1/ (n 3)$ with the correct constant after the Jacobian.

Exercise 8 (hard, short-answer).

Explain in one paragraph why Rademacher's series for $p (n)$ converges while the Hardy-Ramanujan series only provides an asymptotic expansion, referring to the role of the Bessel function $I_{3/2}$ and the Ford circles.

Hint

Hardy-Ramanujan approximate the major-arc integral by a single exponential whose error terms grow; Rademacher integrate over exact Ford circles tangent to the real axis and get an integral that evaluates to $I_{3/2}$ exactly, with a tail that is provably summable.

Answer

Hardy and Ramanujan replaced each major-arc integral by its saddle-point exponential approximation; the resulting series in $k$ has terms whose individual errors, when summed, eventually grow, so the series is asymptotic and divergent — optimal truncation lands within $O (1)$ of $p (n)$ . Rademacher instead parametrised the contour by Ford circles, the circles in the upper half-plane tangent to the real axis at each rational $h / k$ with radius $1/ (2 k^{2})$ ; integrating the transformed generating function over the exact arc of each Ford circle yields, after the change of variables, not an approximate exponential but the modified Bessel function $I_{3/2} (\frac{π}{k} \frac{2}{3} (n - \frac{1}{24}))$ exactly, equivalently the derivative-of- $sinh$ form. Because $I_{3/2} (x) \sim e^{x} / 2 π x$ but the argument carries the factor $1/ k$ , the $k$ -th term decays like $e^{C n / k} / (poly)$ and the tail beyond $k > n$ is rigorously bounded, so the series converges to the exact integer $p (n)$ — rounding the truncated sum recovers $p (n)$ for modest $k$ . Rubric: full credit for contrasting saddle-point approximation (asymptotic) with exact Ford-circle integration (Bessel, convergent) and naming the $1/ k$ decay of the argument.

Advanced results Master

The leading asymptotic is the first term of an exact convergent series, and the same circle-method skeleton governs additive problems far beyond partitions. We collect the Rademacher formula, the role of the eta transformation and Dedekind sums, the general major/minor-arc dissection with its singular series, and the comparison with Ramanujan's tau and the Bessel-function structure.

Theorem 1 (Rademacher's exact convergent series). For $n \geq 1$ ^{[Rademacher 1937]}, $$ p(n) = \frac{1}{\pi\sqrt2}\sum_{k=1}^{\infty} A_k(n),\sqrt k;\frac{d}{dn}!\left(\frac{\sinh!\big(\frac{\pi}{k}\sqrt{\tfrac{2}{3}(n-\tfrac{1}{24})}\big)}{\sqrt{n-\tfrac{1}{24}}}\right), $$ where $A_{k} (n) = \sum_{g c d (h, k) = 1} ω_{h, k} e^{- 2 π inh / k}$ and $ω_{h, k} = e^{π i s (h, k)}$ . The series converges, and the partial sum to $k = O (n)$ rounds to the exact integer $p (n)$ . The $k = 1$ term reproduces the Hardy-Ramanujan asymptotic, since $A_{1} (n) = 1$ and $sinh x / \cdot$ leads with $e^{π 2 n /3} / (4 n 3)$ .

Theorem 2 (eta transformation and the multiplier). Under $τ \mapsto \frac{a τ + b}{c τ + d}$ with $(c d a b) \in SL_{2} (Z)$ , $c > 0$ , $$ \eta!\Big(\frac{a\tau+b}{c\tau+d}\Big) = \varepsilon(a,b,c,d),{-i(c\tau+d)}^{1/2},\eta(\tau), \qquad \varepsilon = \exp!\Big(\pi i\Big(\frac{a+d}{12c} - s(d,c) - \tfrac14\Big)\Big), $$ with $s (d, c)$ the Dedekind sum. The reciprocity law $s (h, k) + s (k, h) = - \frac{1}{4} + \frac{1}{12} (\frac{h}{k} + \frac{k}{h} + \frac{1}{hk})$ makes the multiplier system computable and is what lets $A_{k} (n)$ be evaluated. This transformation, dual to the theta inversion of 21.04.04 and rooted in the Poisson summation of 21.15.01, is the analytic engine of the whole method ^{[Apostol 1990]}.

Theorem 3 (the general circle method; singular series). For an additive problem $r (n) = # {n = x_{1}^{k} + \dots + x_{s}^{k}}$ one writes $r (n) = \int_{0}^{1} f (α)^{s} e (- n α) d α$ with $f (α) = \sum_{x \leq P} e (α x^{k})$ . Dissecting $[0, 1]$ into major arcs $M$ about each $a / q$ with $q \leq P^{δ}$ and minor arcs $m$ for the rest, the major arcs yield the main term ^{[Vaughan 1997]} $$ \int_{\mathfrak{M}} f^s e(-n\alpha),d\alpha \sim \mathfrak{S}(n),\frac{\Gamma(1+1/k)^s}{\Gamma(s/k)},n^{s/k-1}, \qquad \mathfrak{S}(n) = \sum_{q=1}^{\infty}\sum_{\substack{a=1\\gcd(a,q)=1}}^{q}\Big(\frac{S(a,q)}{q}\Big)^s e!\Big(!-\frac{na}{q}\Big), $$ with $S (a, q) = \sum_{x = 0}^{q - 1} e (a x^{k} / q)$ the complete Gauss-Weyl sum. The singular series $S (n)$ encodes the local densities (a product over primes of solution counts mod $p^{j}$ ), and it is exactly the additive analogue of the sum $A_{k} (n)$ in the partition problem. The minor arcs are bounded below the main term by Weyl-sum / Vinogradov estimates; Waring's problem (every large $n$ a sum of $s = s (k)$ $k$ -th powers) and the ternary Goldbach theorem (every large odd $n$ a sum of three primes) are the canonical successes.

Theorem 4 (Bessel-function structure and the index $3/2$ ). The modified Bessel function $I_{ν}$ with $ν = 3/2$ governs the partition formula because the relevant cusp weight is $- 1/2$ : the contour integral $\frac{1}{2 π i} \int_{c - i \infty}^{c + i \infty} t^{- ν - 1} e^{t + x^{2} / (4 t)} d t = x^{- ν} I_{ν} (x)$ produces $I_{3/2}$ , which has the closed form $I_{3/2} (x) = \frac{2}{π x} (cosh x - \frac{s i n h x}{x})$ , equivalently the $\frac{d}{d n} (sinh / \cdot)$ of Theorem 1. The exponential growth $e^{x}$ of $I_{3/2}$ at large argument is precisely the $exp (π 2 n /3 / k)$ of each cusp; the Gamma function of 06.01.15 enters as $Γ (ν + \frac{1}{2})$ in the Bessel normalisation.

Theorem 5 (sharpness and the error term). Truncating Rademacher's series at $N = α n$ terms gives an error $O (n^{- 1/4} exp (π 2 n /3 / (2 N) \cdot c))$ for an explicit constant; Lehmer showed the full series does not converge absolutely term-by-term in a naive bound, yet converges conditionally to $p (n)$ . The same precision is unavailable for Waring's problem, where only an asymptotic with a power-saving error is known and the singular series may vanish for sporadic $n$ (the local obstructions), marking the structural difference between a modular generating function (exact) and a non-modular one (asymptotic only).

Synthesis. Putting these together, the foundational reason the partition problem admits an exact convergent formula while Waring and Goldbach yield only asymptotics is modularity: the partition generating function is $q^{1/24} / η (τ)$ , a weakly holomorphic modular form, so the transformation at every cusp is exact and the cusp contributions assemble — through the Kloosterman-type sums $A_{k} (n)$ and the Bessel function $I_{3/2}$ — into a convergent series, whereas the Weyl sums $f (α) = \sum e (α x^{k})$ of the general method are not modular and only their major arcs admit a main term, leaving genuine minor-arc error. The central insight is that the singular series $S (n)$ of 21.16.01's successor problems is exactly the additive analogue of $A_{k} (n)$ : both are sums over reduced residues weighted by local data, both encode the arithmetic of each cusp, and both descend from the modular or Poisson transformation that this is exactly the content of 21.15.01. This generalises the elementary recurrence of 21.16.01 into an analytic machine and is dual to the multiplicative Euler-product story of 21.04.04; the bridge is that the same contour-and-cusp anatomy — Cauchy integral, Farey/major-arc dissection, transformation at each rational point, saddle or Bessel evaluation — recurs from partitions to Waring to the Goldbach problem, the circle method being one technique wearing many additive disguises.

Full proof set Master

Proposition 1 (the saddle is at $z_{0} = (24 n)^{- 1/2}$ and the leading exponent is $π 2 n /3$ ). The function $g (z) = \frac{π}{12 z} + 2 π n z$ on $z > 0$ attains its minimum at $z_{0} = (24 n)^{- 1/2}$ with $g (z_{0}) = π 2 n /3$ .

Proof. $g^{'} (z) = - \frac{π}{12 z ^{2}} + 2 π n$ , vanishing where $z^{2} = \frac{1}{24 n}$ , i.e. $z_{0} = (24 n)^{- 1/2}$ ; since $g^{''} (z) = \frac{π}{6 z ^{3}} > 0$ this is the unique minimum on $(0, \infty)$ . Substituting, $g (z_{0}) = \frac{π}{12} 24 n + 2 π n (24 n)^{- 1/2} = \frac{π 24 n}{12} + \frac{2 π n}{24 n}$ . Writing $24 n = 26 n$ , the first term is $\frac{π \cdot 2 6 n}{12} = \frac{π 6 n}{6} = \frac{π n}{6}$ and the second is $\frac{2 π n}{2 6 n} = \frac{π n}{6}$ ; their sum is $\frac{2 π n}{6} = 2 π n /6 = π 4 n /6 = π 2 n /3$ . $□$

Proposition 2 (the $k = 1$ Rademacher term is the Hardy-Ramanujan asymptotic). With $A_{1} (n) = 1$ , the $k = 1$ term of Theorem 1 is asymptotic to $\frac{1}{4 n 3} e^{π 2 n /3}$ as $n \to \infty$ .

Proof. The $k = 1$ term is $\frac{1}{π 2} \frac{d}{d n} (\frac{s i n h ( π ( 2/3 ) ( n - 1/24 ) )}{n - 1/24})$ . Put $μ = π (2/3) (n - 1/24)$ , so $sinh μ \sim \frac{1}{2} e^{μ}$ for large $n$ and $μ \sim π 2 n /3$ . Differentiating, the dominant contribution comes from differentiating the exponential: $\frac{d μ}{d n} = \frac{π}{2} 2/3 (n - 1/24)^{- 1/2} \sim \frac{π}{6} \frac{1}{2 n} \cdot ?$ , and combined with the prefactor $\frac{1}{n - 1/24} \sim n^{- 1/2}$ , the derivative yields $\frac{1}{π 2} \cdot \frac{1}{2} e^{μ} \cdot \frac{d μ}{d n} \cdot n^{- 1/2} (1 + o (1))$ . Collecting the algebraic factors: $\frac{1}{π 2} \cdot \frac{1}{2} \cdot \frac{π}{6} \cdot n^{- 1/2} \cdot n^{- 1/2} = \frac{1}{2 2 6 n} = \frac{1}{4 3 n}$ , since $26 = 12 = 23$ . Hence the $k = 1$ term is $\frac{1}{4 n 3} e^{π 2 n /3} (1 + o (1))$ . $□$

Proposition 3 (Dedekind reciprocity makes $A_{k} (n)$ computable). The Dedekind sum satisfies $s (h, k) + s (k, h) = - \frac{1}{4} + \frac{1}{12} (\frac{h}{k} + \frac{k}{h} + \frac{1}{hk})$ for $g cd (h, k) = 1$ , and consequently each multiplier $ω_{h, k} = e^{π i s (h, k)}$ is a computable root of unity times a controlled phase.

Proof. Reciprocity is the standard finite-Fourier evaluation of the sawtooth sum $\sum_{r} ((r / k)) ((h r / k))$ ; it follows from the cotangent representation $s (h, k) = \frac{1}{4 k} \sum_{r = 1}^{k - 1} cot \frac{π r}{k} cot \frac{π h r}{k}$ and the partial-fraction identity for $cot$ applied symmetrically in $h, k$ ^{[Apostol 1990]}. Given reciprocity, $s (h, k)$ is determined recursively by a Euclidean descent on $(h, k)$ exactly like a continued-fraction expansion, terminating because $g cd$ strictly decreases; each step adds an explicit rational, so $s (h, k) \in Q$ with denominator dividing $12 k$ . Therefore $ω_{h, k} = e^{π i s (h, k)}$ is a $24 k$ -th root of unity, and the finite sum $A_{k} (n) = \sum_{g c d (h, k) = 1} ω_{h, k} e^{- 2 π inh / k}$ is a finite sum of explicit roots of unity, hence computable in closed form for each $k$ . $□$

Proposition 4 (major/minor split: the main term comes only from major arcs). In $r (n) = \int_{0}^{1} f (α)^{s} e (- n α) d α$ with $f (α) = \sum_{x \leq P} e (α x^{k})$ and $s$ large enough, $\int_{m} f^{s} e (- n α) d α = o (n^{s / k - 1})$ , so the asymptotic is determined by the major arcs.

Proof sketch (the structure, with the Weyl input cited). On a minor arc $α$ is poorly approximable: $∣ α - a / q ∣ \leq q^{- 2}$ forces $P^{δ} < q \leq P^{k - δ}$ . Weyl's inequality (or Vinogradov's mean value for large $k$ ) gives $sup_{α \in m} ∣ f (α) ∣ ≪ P^{1 - σ}$ for an explicit $σ = σ (k) > 0$ ^{[Vaughan 1997]}. Then $\int_{m} ∣ f ∣^{s} \leq (sup_{m} ∣ f ∣)^{s - t} \int_{0}^{1} ∣ f ∣^{t} d α$ for a suitable even $t$ where $\int_{0}^{1} ∣ f ∣^{t}$ counts solutions of a $t$ -fold equation and is bounded by $P^{t - k + ε}$ (the mean-value estimate). Choosing $s$ large enough that $(1 - σ) (s - t) + (t - k) < s / k - 1$ makes the minor-arc total $o (n^{s / k - 1})$ . The major arcs then give the main term of Theorem 3 by the Gauss-sum/singular-integral factorisation. $□$

Connections Master

The asymptotic and exact formulas of this unit are the analytic completion of the elementary partition theory of 21.16.01: the pentagonal recurrence pins $p (n)$ down exactly but opaquely, while the circle method reads off its size and an exact convergent series; both are governed by the same Euler function $(q; q)_{\infty} = q^{- 1/24} η (τ)$ , so the combinatorial identity and the modular transformation are two faces of one object.
The modular transformation $η (- 1/ τ) = - i τ η (τ)$ that drives every cusp expansion here is proved by the Poisson summation formula of 21.15.01; the saddle-point/Laplace estimate of the major-arc integral is the additive-number-theory instance of the stationary-phase and Voronoi-summation machinery developed there.
The Bessel-function and Gamma-function structure of Rademacher's series rests on the contour representation of $1/Γ$ and the special-function identities of 06.01.15; the index $ν = 3/2$ , the closed form $I_{3/2} (x) = 2/ (π x) (cosh x - sinh x / x)$ , and the Mellin-Barnes integral all live in the Gamma-function toolkit.
The Jacobi triple product and theta inversion of 21.04.04 are the multiplicative-modular siblings of the partition story: the singular series $S (n)$ of Waring's problem is the additive analogue of the Euler product, and the eta transformation used here is dual to the theta transformation proved there, so partitions, sums of squares, and Waring's problem all descend from the same modular transformation law.

Historical & philosophical context Master

The circle method was created in Hardy and Ramanujan's 1918 paper ^{[Hardy-Ramanujan 1918]} to determine the growth of $p (n)$ . Ramanujan's intuition, sharpened by Hardy's analysis, was to integrate the generating function over a circle approaching the unit circle and to localise the integral near the roots of unity, where the modular transformation of the eta function controls the singularity. Their result was the asymptotic $p (n) \sim e^{π 2 n /3} / (4 n 3)$ together with a finite asymptotic series that, optimally truncated, determined $p (n)$ exactly for the values they could check — a feat that astonished MacMahon, who had computed $p (n)$ up to $n = 200$ by the pentagonal recurrence.

Rademacher's 1937 paper ^{[Rademacher 1937]} removed the asymptotic-divergence defect by replacing the Hardy-Ramanujan circular arcs with the Ford circles tangent to the real axis at each rational, and the dominant exponential by the modified Bessel function $I_{3/2}$ ; the outcome was the first exactly convergent series for an additive arithmetic function. In the hands of Hardy and Littlewood through the 1920s the same dissection — recast as the major and minor arcs of the unit interval with the singular series as the arithmetic main term — became the Hardy-Littlewood method, the standard tool for Waring's problem and, via Vinogradov's 1937 minor-arc estimates, for the ternary Goldbach theorem ^{[Vaughan 1997]}. The Dedekind sums $s (h, k)$ that index the partition multipliers, introduced by Dedekind in his 1877 commentary on Riemann's fragments on elliptic functions and governed by the reciprocity law, recur throughout, tying the partition asymptotics to the transformation theory of modular forms developed by Apostol and others ^{[Apostol 1990]}.

Bibliography Master

@article{hardyramanujan1918circle,
  author  = {Hardy, G. H. and Ramanujan, S.},
  title   = {Asymptotic formulae in combinatory analysis},
  journal = {Proceedings of the London Mathematical Society (2)},
  volume  = {17},
  pages   = {75--115},
  year    = {1918}
}

@article{rademacher1937,
  author  = {Rademacher, Hans},
  title   = {On the partition function $p(n)$},
  journal = {Proceedings of the London Mathematical Society (2)},
  volume  = {43},
  pages   = {241--254},
  year    = {1937}
}

@book{apostol1990modular,
  author    = {Apostol, Tom M.},
  title     = {Modular Functions and Dirichlet Series in Number Theory},
  series    = {Graduate Texts in Mathematics},
  volume    = {41},
  edition   = {2nd},
  publisher = {Springer-Verlag},
  year      = {1990}
}

@book{vaughan1997hardylittlewood,
  author    = {Vaughan, R. C.},
  title     = {The Hardy-Littlewood Method},
  series    = {Cambridge Tracts in Mathematics},
  volume    = {125},
  edition   = {2nd},
  publisher = {Cambridge University Press},
  year      = {1997}
}

@book{andrews1976partitions,
  author    = {Andrews, George E.},
  title     = {The Theory of Partitions},
  series    = {Encyclopedia of Mathematics and its Applications},
  volume    = {2},
  publisher = {Addison-Wesley},
  year      = {1976}
}

@article{rademacher1938zuckerman,
  author  = {Rademacher, Hans and Zuckerman, Herbert S.},
  title   = {On the Fourier coefficients of certain modular forms of positive dimension},
  journal = {Annals of Mathematics (2)},
  volume  = {39},
  number  = {2},
  pages   = {433--462},
  year    = {1938}
}

@article{lehmer1937series,
  author  = {Lehmer, D. H.},
  title   = {On the series for the partition function},
  journal = {Transactions of the American Mathematical Society},
  volume  = {43},
  number  = {2},
  pages   = {271--295},
  year    = {1938}
}

Prerequisites

21.16.01
21.15.01
06.01.15

Tier anchors

beginner: Andrews-Eriksson 2004 *Integer Partitions* (Cambridge) Ch. 1, 5 (the growth of $p(n)$ and where the exponential comes from); Marcus du Sautoy / Mathologer 'The strange numbers that count partitions' for the size of $p(n)$ visualised; Newman 1998 *Analytic Number Theory* (Springer GTM 177) Ch. 16 for the elementary contour heuristic
intermediate: Apostol 1990 *Modular Functions and Dirichlet Series in Number Theory* (Springer GTM 41, 2nd ed.) Ch. 5 (the Rademacher convergent series for $p(n)$, the Farey dissection, the transformation of $\eta$); Apostol 1976 *Introduction to Analytic Number Theory* (Springer UTM) Ch. 14 §14.10 (the Hardy-Ramanujan asymptotic stated)
master: Apostol 1990 *Modular Functions and Dirichlet Series in Number Theory* (Springer GTM 41) Ch. 3 (Dedekind eta and Dedekind sums), Ch. 5 (the Rademacher exact formula via the circle method); Hardy-Ramanujan 1918 *Proc. London Math. Soc.* (2) 17, 75-115 (the original circle method and asymptotic); Rademacher 1937 *Proc. London Math. Soc.* (2) 43, 241-254 (the convergent series); Vaughan 1997 *The Hardy-Littlewood Method* (Cambridge Tracts 125, 2nd ed.) Ch. 1-2 (the circle method for Waring and Goldbach, major/minor arcs, the singular series); Andrews 1976 *The Theory of Partitions* (Addison-Wesley) Ch. 5

References

Apostol, T. M. — Modular Functions and Dirichlet Series in Number Theory · Springer Graduate Texts in Mathematics 41, 2nd edition (1990). Chapter 3 develops the Dedekind eta function and Dedekind sums $s(h,k)$ with the reciprocity law; Chapter 5 ('Rademacher's series for the partition function') gives the full circle-method derivation: the Cauchy integral $p(n)=\frac{1}{2\pi i}\oint F(q)q^{-n-1}dq$, the Farey dissection of the unit circle into arcs about $e^{2\pi i h/k}$, the modular transformation of $F(q)=1/((q;q)_\infty)$ inherited from $\eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau)$, the Kloosterman-type sums $A_k(n)=\sum_{h}\omega_{h,k}e^{-2\pi i nh/k}$ built from $\eta$-multipliers, and the Bessel-function $I_{3/2}$ main terms that sum to the exact convergent series.
Hardy, G. H. & Ramanujan, S. — Asymptotic formulae in combinatory analysis · *Proceedings of the London Mathematical Society* (2) 17 (1918), 75-115. The paper that introduced the circle method: the generating function $F(q)=\prod(1-q^n)^{-1}$ is integrated over a circle of radius approaching $1$, the dominant contribution coming from the neighbourhood of $q=1$ (and lesser ones from $q=e^{2\pi i h/k}$), yielding $p(n)\sim e^{\pi\sqrt{2n/3}}/(4n\sqrt3)$ together with an asymptotic (divergent) series of corrections.
Rademacher, H. — On the partition function p(n) · *Proceedings of the London Mathematical Society* (2) 43 (1937), 241-254. Rademacher replaced the Hardy-Ramanujan circular arcs by Ford circles in the upper half-plane and the divergent tail by the Bessel function $I_{3/2}$, converting the asymptotic series into an exactly convergent series $p(n)=\frac{1}{\pi\sqrt2}\sum_{k\geq1}A_k(n)\sqrt k\,\frac{d}{dn}\frac{\sinh((\pi/k)\sqrt{(2/3)(n-1/24)})}{\sqrt{n-1/24}}$, the first exact formula for an additive arithmetic function of this type.
Vaughan, R. C. — The Hardy-Littlewood Method · Cambridge Tracts in Mathematics 125, 2nd edition (1997). The circle method as a general technique in additive number theory: the major/minor-arc dissection of the unit circle, the singular series $\mathfrak{S}(n)=\sum_{q}\sum_{a}^{*}(S(a/q)/q)^s e(-na/q)$, the major-arc main term for Waring's problem $r_{k,s}(n)\sim \mathfrak{S}(n)\Gamma(1+1/k)^s\Gamma(s/k)^{-1}n^{s/k-1}$, and the minor-arc estimates (Weyl sums, Vinogradov's method) needed to bound the error.
Andrews, G. E. — The Theory of Partitions · Addison-Wesley, Encyclopedia of Mathematics and its Applications, Vol. 2 (1976; Cambridge reissue 1998). Chapter 5 surveys the asymptotics of $p(n)$, the Hardy-Ramanujan-Rademacher series, and the role of the modular transformation of the generating function in extracting the leading exponential growth.

Estimated time

beginner: 18m
intermediate: 45m
master: 90m