8 Proof of Theorem 5.2

Our proof of the Theorem 5.2 relies on the following two inequalities for modular objects.

Proposition 8.1

Consider the function $A : (0, \infty) \to C$ defined as

A (t) := - t^{2} ϕ_{0} (i / t) - \frac{36}{π^{2}} ψ_{I} (i t) .

183

Then

A (t) < 0

184

for all $t > 0$ .

Proposition 8.2

Consider the function $B : (0, \infty) \to C$ defined as

B (t) := - t^{2} ϕ_{0} (i / t) + \frac{36}{π^{2}} ψ_{I} (i t)

185

Then

B (t) > 0

186

for all $t > 0$ .

Here we formalize the proof of the inequalities by Lee [ 7 ] . First, we can rewrite the inequality in 8.1 as follows.

Definition 8.3

Define two (quasi) modular forms as

\begin{aligned} F (z) & = (E_{2} (z) E_{4} (z) - E_{6} (z))^{2} \\ G (z) & = H_{2} (z)^{3} (2 H_{2} (z)^{2} + 5 H_{2} (z) H_{4} (z) + 5 H_{4} (z)^{2}) . \end{aligned}

Lemma 8.4

We have

\begin{aligned} ϕ_{0} & = \frac{F}{Δ} \\ ψ_{S} & = - \frac{1}{2} \frac{G}{Δ} \end{aligned}

Proof ▶

Lemma 8.5

Inequality 184 and 186 are equivalent to

\begin{array}{r} F (i t) + \frac{18}{π^{2}} G (i t) > 0 \\ F (i t) - \frac{18}{π^{2}} G (i t) > 0 \end{array}

respectively.

Proof ▶

By 151,

ψ_{I} (i t) = (ψ_{S} |_{- 2} S) (i t) = (i t)^{2} ψ_{S} (- \frac{1}{i t}) = - t^{2} ψ_{S} (\frac{i}{t}) .

193

Combined with Lemma 8.4 we can rewrite 184 as

A (t) = - t^{2} ϕ_{0} (\frac{i}{t}) + \frac{36}{π^{2}} ψ_{S} (\frac{i}{t}) < 0 \Leftrightarrow \frac{F (i t)}{Δ (i t)} + \frac{18}{π^{2}} \frac{G (i t)}{Δ (i t)} > 0

194

for $t > 0$ , which is equivalent to 191 by Corollary 6.25. Equivalences of 186 and 192 follows similarly; just change the sign.

Now, the first inequality 191 follows from the positivity of each $F (i t)$ and $G (i t)$ .

Lemma 8.6

For all $t > 0$ , we have $F (i t) > 0$ and $G (i t) > 0$ .

Proof ▶

By Ramanujan’s identity 77, we have $F (z) = 9 E_{4}^{'} (z)^{2}$ and

F (i t) = 9 E_{4}^{'} (i t)^{2} = 9 {(240 \sum_{n \geq 1} n σ_{3} (n) e^{- 2 π n t})}^{2} > 0.

195

$G (i t) > 0$ follows from positivity of $H_{2} (i t)$ and $H_{4} (i t)$ (Lemma 6.42).

Corollary 8.7

191 holds.

Proof ▶

To prove the second inequality 192, we need some identities satisfied by $F$ and $G$ .

Lemma 8.8

$F$ and $G$ satisfy the following differential equations:

\begin{aligned} \partial_{12} \partial_{10} F - \frac{5}{6} E_{4} F & = 7200 Δ (- E_{2}^{'}) \\ \partial_{12} \partial_{10} G - \frac{5}{6} E_{4} G & = - 640 Δ H_{2} \end{aligned}

Proof ▶

Both can be shown by direct computations. By Ramanujan’s identities (Theorem 6.48) and the product rule of Serre derivatives (Theorem 6.51), we have

\begin{aligned} \partial_{5} (E_{2} E_{4} - E_{6}) & = (E_{2} E_{4} - E_{6})^{'} - \frac{5}{12} E_{2} (E_{2} E_{4} - E_{6}) \\ = \frac{E_{2}^{2} - E_{4}}{12} \cdot E_{4} + E_{2} \cdot \frac{E_{2} E_{4} - E_{6}}{3} - \frac{E_{2} E_{6} - E_{4}^{2}}{2} - \frac{5}{12} E_{2} (E_{2} E_{4} - E_{6}) \\ = - \frac{5}{12} (E_{2} E_{6} - E_{4}^{2}) \\ \partial_{7} (E_{2} E_{6} - E_{4}^{2}) & = (E_{2} E_{6} - E_{4}^{2})^{'} - \frac{7}{12} E_{2} (E_{2} E_{6} - E_{4}^{2}) \\ = \frac{E_{2}^{2} - E_{4}}{12} \cdot E_{6} + E_{2} \cdot \frac{E_{2} E_{6} - E_{4}^{2}}{2} - 2 E_{4} \cdot \frac{E_{2} E_{4} - E_{6}}{3} - \frac{7}{12} E_{2} (E_{2} E_{6} - E_{4}^{2}) \\ = - \frac{7}{12} E_{4} (E_{2} E_{4} - E_{6}) \end{aligned}

and using these we can compute

\begin{aligned} \partial_{10} F & = \partial_{10} (E_{2} E_{4} - E_{6})^{2} \\ = 2 (E_{2} E_{4} - E_{6}) \partial_{5} (E_{2} E_{4} - E_{6}) \\ = - \frac{6}{5} (E_{2} E_{4} - E_{6}) (E_{2} E_{6} - E_{4}^{2}), \\ \partial_{12} \partial_{10} F & = - \frac{5}{6} \partial_{12} ((E_{2} E_{4} - E_{6}) (E_{2} E_{6} - E_{4})) \\ = - \frac{5}{6} (\partial_{5} (E_{2} E_{4} - E_{6})) (E_{2} E_{6} - E_{4}^{2}) - \frac{5}{6} (E_{2} E_{4} - E_{6}) (\partial_{7} (E_{2} E_{6} - E_{4})) \\ = \frac{25}{72} (E_{2} E_{6} - E_{4}^{2})^{2} + \frac{35}{72} E_{4} (E_{2} E_{4} - E_{6})^{2}, \\ \partial_{12} \partial_{10} F - \frac{5}{6} E_{4} F & = \frac{25}{72} (E_{2} E_{6} - E_{4}^{2})^{2} + \frac{35}{72} E_{4} (E_{2} E_{4} - E_{6})^{2} - \frac{5}{6} E_{4} (E_{2} E_{4} - E_{6})^{2} \\ = \frac{25}{72} ((E_{2} E_{6} - E_{4}^{2})^{2} - E_{4} (E_{2} E_{4} - E_{6})^{2}) \\ = \frac{25}{72} (- E_{2}^{2} E_{4}^{3} + E_{2}^{2} E_{6}^{2} + E_{4}^{4} - E_{4} E_{6}^{3}) \\ = - \frac{25}{72} (E_{4}^{3} - E_{6}^{2}) (E_{2}^{2} - E_{4}) \\ = 7200 \cdot \frac{E_{4}^{3} - E_{6}^{2}}{1728} \cdot \frac{- E_{2}^{2} + E_{4}}{12} \\ = 7200 Δ (- E_{2}^{'}) \end{aligned}

which proves 196. Similarly, 197 can be proved using Proposition 6.50 and Lemma 6.41.

Corollary 8.9

196 (resp. 197) is positive (resp. negative) on the (positive) imaginary axis.

Proof ▶

From 13 and Lemma 6.25,

7200 (- E_{2}^{'} (i t)) Δ (i t) = 7200 \cdot 24 (\sum_{n \geq 1} n σ_{1} (n) e^{- 2 π n t}) \cdot Δ (i t) > 0.

216

Negativity of 197, i.e. $- 640 Δ (i t) H_{2} (i t) < 0$ follows from Corollary 6.42 and 6.25.

The second inequality 192 follows from the following two observations. Since $G (i t) > 0$ for all $t > 0$ , we can define the quotient

Q (t) := \frac{F (i t)}{G (i t)}

216

as a function on $(0, \infty)$ .

Lemma 8.10

We have

lim_{t \to 0^{+}} Q (t) = \frac{18}{π^{2}} .

217

Proof ▶

We have

lim_{t \to 0^{+}} Q (t) = lim_{t \to 0^{+}} \frac{F (i t)}{G (i t)} = lim_{t \to \infty} \frac{F (i / t)}{G (i / t)} .

218

By using the transformation laws of Eisenstein series 14, 10 (for $k = 4, 6$ ) and the thetanull functions, 28, 30, we get

\begin{aligned} F (\frac{i}{t}) & = t^{12} F (i t) - \frac{12 t^{11}}{π} (E_{2} (i t) E_{4} (i t) - E_{6} (i t)) E_{4} (i t) + \frac{36 t^{10}}{π^{2}} E_{4} (i t)^{2}, \\ G (\frac{i}{t}) & = t^{10} H_{4} (i t)^{3} (2 H_{4} (i t)^{2} + 5 H_{4} (i t) H_{2} (i t) + 5 H_{2} (i t)^{2}) . \end{aligned}

Since $F$ , $E_{2} E_{4} - E_{6}$ and $H_{2}$ are cusp forms, we have $lim_{t \to \infty} t^{k} A (i t) = 0$ when $A (z)$ is one of these forms and $k \geq 0$ . From $lim_{t \to \infty} E_{4} (i t) = 1 = lim_{t \to \infty} H_{4} (i t)$ , we get

\begin{aligned} lim_{t \to \infty} \frac{F (i / t)}{G (i / t)} & = lim_{t \to \infty} \frac{t^{12} F (i t) - \frac{12 t^{11}}{π} (E_{2} (i t) E_{4} (i t) - E_{6} (i t)) E_{4} (i t) + \frac{36 t^{10}}{π^{2}} E_{4} (i t)^{2}}{t^{10} H_{4} (i t)^{3} (2 H_{4} (i t)^{2} + 5 H_{4} (i t) H_{2} (i t) + 5 H_{2} (i t)^{2})} \\ = lim_{t \to \infty} \frac{t^{2} F (i t) - \frac{12 t}{π} (E_{2} (i t) E_{4} (i t) - E_{6} (i t)) E_{4} (i t) + \frac{36}{π^{2}} E_{4} (i t)^{2}}{H_{4} (i t)^{3} (2 H_{4} (i t)^{2} + 5 H_{4} (i t) H_{2} (i t) + 5 H_{2} (i t)^{2})} \\ = \frac{18}{π^{2}} . \end{aligned}

Proposition 8.11

The function $t \mapsto Q (t)$ is monotone decreasing.

Proof ▶

It is enough to show that

\begin{aligned} \frac{d}{d t} (\frac{F (i t)}{G (i t)}) < 0 & \Leftrightarrow (- 2 π) \frac{F^{'} (i t) G (i t) - F (i t) G^{'} (i t)}{G (i t)^{2}} < 0 \\ \Leftrightarrow F^{'} (i t) G (i t) - F (i t) G^{'} (i t) > 0 \\ \Leftrightarrow (\partial_{10} F) (i t) G (i t) - F (i t) (\partial_{10} G) (i t) > 0. \end{aligned}

Let $L_{1, 0} := (\partial_{10} F) G - F (\partial_{10} G)$ . Then its Fourier expansion starts with

L_{1, 0} = 5308416000 q^{\frac{7}{2}} + O (q^{\frac{9}{2}})

227

and its Serre derivative $\partial_{22} L_{1, 0}$ is positive by Corollary 8.9:

\begin{array}{r} \partial_{22} L_{1, 0} = (\partial_{12} \partial_{10} F) G - F (\partial_{12} \partial_{10} G) = Δ (7200 (- E_{2}^{'}) G + 640 H_{2} F) > 0. \end{array}

Hence $L_{1, 0} (i t) > 0$ by Theorem 6.52, and the monotonicity follows.

Corollary 8.12

192 holds.

Proof ▶

\frac{F (i t)}{G (i t)} = Q (t) < lim_{u \to 0^{+}} Q (u) = \frac{18}{π^{2}}

228

and by Lemma 8.6, 192 follows.

Finally, we are ready to prove Theorem 5.2.

Theorem 8.13

The function

g (x) := \frac{π i}{8640} a (x) + \frac{i}{240 π} b (x)

satisfies conditions 6–8.

Proof ▶

First, we prove that 6 holds. By Propositions 7.11 and 7.25 we know that for $r > \sqrt{2}$

g (r) = \frac{π}{2160} \sin (π r^{2} / 2)^{2} \int_{0}^{\infty} A (t) e^{- π r^{2} t} d t

229

where

A (t) = - t^{2} ϕ_{0} (i / t) - \frac{36}{π^{2}} ψ_{I} (i t) .

from the Proposition 8.1 we know that $A (t) < 0 for t \in (0, \infty) .$ Therefore identity 229 implies 6.

Next, we prove 7. By Propositions 7.12 and 7.26 we know that for $r > 0$

\hat{g} (r) = \frac{π}{2160} \sin (π r^{2} / 2)^{2} \int_{0}^{\infty} B (t) e^{- π r^{2} t} d t

230

where

B (t) = - t^{2} ϕ_{0} (i / t) + \frac{36}{π^{2}} ψ_{I} (i t) .

Finally, the property 8 readily follows from Proposition 7.13 and Proposition 7.27. This finishes the proof of Theorems 8.13 and 5.2. □

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Email address: maryna.viazovska@epfl.ch