For n ≥ 0 n\geq 0 n ≥ 0 ( x ; q ) n = ( 1 − x ) ( 1 − x q ) ⋯ ( 1 − x q n − 1 ) (x;q)_n=(1-x)(1-xq)\cdots(1-xq^{n-1}) ( x ; q ) n = ( 1 − x ) ( 1 − x q ) ⋯ ( 1 − x q n − 1 ) q q q [ n k ] q \begin{bmatrix}n\\k\end{bmatrix}_q [ n k ] q q q q m ≥ 2 m\geq 2 m ≥ 2 0 ≤ a ≤ m − 2 0\leq a\leq m-2 0 ≤ a ≤ m − 2
Φ ~ m ( a ) ( q ) = m q ( m − a − 1 ) 2 4 m ∑ n 1 , … , n m − 1 ≥ 0 ( − 1 ) n m − 1 q ( n m − 1 + 1 2 ) + n 1 2 + ⋯ + n m − 2 2 + n a + 1 + ⋯ + n m − 2 ∏ i = 1 m − 2 [ n i + 1 + δ i , a i ] q . \widetilde{\Phi}_m^{(a)}(q)=mq^{\frac{(m-a-1)^2}{4m}}\sum_{n_1,\dots,n_{m-1}\geq 0}(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_1^2+\cdots+n_{m-2}^2+n_{a+1}+\cdots+n_{m-2}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\
_i\end{bmatrix}_q. Φ m ( a ) ( q ) = m q 4 m ( m − a − 1 ) 2 n 1 , … , n m − 1 ≥ 0 ∑ ( − 1 ) n m − 1 q ( 2 n m − 1 + 1 ) + n 1 2 + ⋯ + n m − 2 2 + n a + 1 + ⋯ + n m − 2 i = 1 ∏ m − 2 [ n i + 1 + δ i , a i ] q . It has the false-theta expansion
Φ ~ m ( a ) ( q ) = m ∑ n ≥ 0 χ 2 m ( a ) ( n ) q n 2 / 4 m , \widetilde{\Phi}_m^{(a)}(q)=m\sum_{n\geq0}\chi_{2m}^{(a)}(n)q^{n^2/4m}, Φ m ( a ) ( q ) = m n ≥ 0 ∑ χ 2 m ( a ) ( n ) q n 2 /4 m , where χ 2 m ( a ) ( n ) = 1 \chi_{2m}^{(a)}(n)=1 χ 2 m ( a ) ( n ) = 1 n ≡ m − a − 1 ( m o d 2 m ) n\equiv m-a-1\pmod{2m} n ≡ m − a − 1 ( mod 2 m ) − 1 -1 − 1 n ≡ m + a + 1 ( m o d 2 m ) n\equiv m+a+1\pmod{2m} n ≡ m + a + 1 ( mod 2 m ) 0 0 0
Y m , N ( a ) ( q ) = ∑ n 1 , … , n m − 1 = 0 N − 1 ( − 1 ) n m − 1 q ( n m − 1 + 1 2 ) + n 1 2 + ⋯ + n m − 2 2 + n a + 1 + ⋯ + n m − 2 ∏ i = 1 m − 2 [ n i + 1 + δ i , a i ] q . Y_{m,N}^{(a)}(q)=\sum_{n_1,\dots,n_{m-1}=0}^{N-1}(-1)^{n_{m-1}}q^{\binom{n_{m-1}+1}{2}+n_1^2+\cdots+n_{m-2}^2+n_{a+1}+\cdots+n_{m-2}}\prod_{i=1}^{m-2}\begin{bmatrix}n_{i+1}+\delta_{i,a}\
_i\end{bmatrix}_q. Y m , N ( a ) ( q ) = n 1 , … , n m − 1 = 0 ∑ N − 1 ( − 1 ) n m − 1 q ( 2 n m − 1 + 1 ) + n 1 2 + ⋯ + n m − 2 2 + n a + 1 + ⋯ + n m − 2 i = 1 ∏ m − 2 [ n i + 1 + δ i , a i ] q . Let ζ N = e 2 π i / N \zeta_N=e^{2\pi i/N} ζ N = e 2 π i / N Hikami's conjecture. For any m ≥ 2 m\geq2 m ≥ 2 0 ≤ a ≤ m − 2 0\leq a\leq m-2 0 ≤ a ≤ m − 2
lim q → ζ N Φ ~ m ( a ) ( q ) = ζ N ( m − a − 1 ) 2 4 m Y m , N ( a ) ( ζ N ) . \lim_{q\to\zeta_N}\widetilde{\Phi}_m^{(a)}(q)=\zeta_N^{\frac{(m-a-1)^2}{4m}}Y_{m,N}^{(a)}(\zeta_N). q → ζ N lim Φ m ( a ) ( q ) = ζ N 4 m ( m − a − 1 ) 2 Y m , N ( a ) ( ζ N ) . False theta functions have well-defined radial limits at roots of unity, and these limits give quantum modular forms. The paper proves this conjecture, so its status is solved.
