CORRECTIONS

• Page 75, equation (2.11). The three ± signs should be removed; the equation should

read

θ [ \begin{matrix}ϵ + 2m \\ ϵ′ + 2n \end{matrix} ] = (exp πıϵn) θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] .

• Page 105, line 10. Replace “characteristic” by “characteristics”.

• Page 111, Table 4. The entries in the last two rows of the table need to be corrected. The correct version is Table 1 below.

• Page 127, line 5. Replace “ℑζ = 1” by “ℑζ = t.”

• Page 194, equation (3.19) should read

θ_{3}^{3} = θ_{2}^{3} − θ_{1}^{3} and θ_{4}^{3} = ϵ^{2}θ_{3}^{3} + ϵθ_{1}^{3},

where ϵ = exp \frac{πı}{3}.

• Page 202. A formula needs correction:

(Φ [ \frac{1}{5} \frac{3}{5} ]) = \frac{P_{2}^{15}P_{− \frac{1}{2}}^{15}}{P_{∞}^{9}P_{0}P_{1}^{9}P_{−1}P_{−2}^{9}P_{\frac{1}{2}} P 9_{\frac{3}{2}}P 9_{\frac{5}{2}}P_{− \frac{3}{2}} P_{\frac{2}{5}}} .

• Page 247, last 4 lines. There are too many identifications of objects on the Riemann surface and their lifts to the universal cover of the surface. It is best to replace these

lines by: fact nothing more than (a nonzero multiple of) (θ′)^{24} [ \begin{matrix}1 \\ 1 \end{matrix} ]. The divisor of

the 6-form W consists of the lifts to ℍ^{2} ∪ℚ∪\{∞\} of the Weierstrass points of \overline{ℍ^{2}/Γ(7)} and hence equals D. The map Φ of \overline{ℍ^{2}/Γ(7)} into Pℂ^{2} is defined by the projections to \overline{ℍ^{2}/Γ(7)} of the two triples (φ_{0}, φ_{1}, φ_{2}) and W^{\frac{1}{8}} (φ_{0}, φ_{1}, φ_{2}). One easily sees that W^{\frac{1}{8}} φ_{l}, l = 0, 1, 2, projects to a holomorphic 1-form

• Pages 265 and 277. In formula (4.1) and Theorem 3.2, one needs to assume that N is an odd positive integer. A more general formula valid for all N ∈ ℤ^{+} and all k ∈ ℤ of the same parity as N is

∑_{l=1}^{N}(−1)^{l}θ^{N} [ \frac{k}{N} ; \frac{2N−2l+1}{N} ] = 0.

• Page 272, line 11. The displayed formula should read

\frac{c_{1}}{2} (θ [ 1^{\frac{1}{3}} ] (3ζ, 3τ) − θ [ 1^{\frac{1}{3}} ] (−3ζ, 3τ)) + c_{2}θ [ \begin{matrix}1 \\ 1 \end{matrix} ] (3ζ, 3τ) = θ^{3} [ \begin{matrix}1 \\ 1 \end{matrix} ] (ζ, τ).

• Page 303. Formula (4.29) should read

x ∏_{n=1}^{∞} \frac{(1 − x^{9n})^{3}(1 − x^{n})^{3}}{(1 − x^{3n})^{2}} = \frac{\sqrt{3}}{3}ı ∑_{n=1}^{∞} (\frac{n(ω^{2}x)^{n}}{1 − (ω^{2}x)^{n}} − \frac{n(ωx)^{n}}{1 − (ωx)^{n}} ) .

• Page 313. The last formula above Corollary 9.2 should read

\frac{d}{dτ} log (\frac{θ [ 1^{\frac{3}{5}} ] (0, 5τ)}{θ [ 1^{\frac{1}{5}} ] (0, 5τ)}) = 2πıx (\frac{1}{5x} − ∑_{n=1}^{∞} \frac{χ(n)nx^{n−1}}{1 − x^{n}} ) .

• Page 320, line 9. “\frac{1}{2} + \frac{\sqrt{3}}{6}ı” should replace “\frac{3}{2} + \frac{\sqrt{3}}{2}ı”.

• Page 320. A typographical error in the formula in Theorem 9.12 should be corrected. The formula should read

\frac{θ^{3} [ 1_{\frac{1}{3}} ]}{θ [ \frac{1}{3} \frac{1}{3} ] θ [ 1^{\frac{1}{3}} ] θ [ \frac{1}{3} \frac{5}{3} ]}(0, τ) = c (\frac{η(3τ)}{η(τ)})^{4} .

• Page 449. In Theorem 2.3, n is restricted to belong to ℤ^{+}.

• Page 513. In the first paragraph “to to the topics” should be replaced by “to the topics.”

• Page 519. The last displayed equation should read

θ′ [ 1^{\frac{2}{3}} ] (0, 6τ) = \frac{θ [ \begin{matrix}1 \\ 0 \end{matrix} ] (0, τ)θ′ [ \begin{matrix}1 \\ 1 \end{matrix} ] (0, 2τ)}{6θ [ 1^{\frac{1}{3}} ] (0, 3τ)}.

• Page 520. The last displayed equation should read

(θ′ [ 1^{\frac{2}{3}} ] (0, 6γ(τ)))^{24} (γ′(τ))^{6} = c_{γ} (θ′ [ 1^{\frac{2}{3}} ] (0, 6τ))^{24} , γ ∈ Γ_{o}(6),

• Page 521. The formula on the second line should read

θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ] (0, τ)θ [ \begin{matrix}δ \\ δ′ \end{matrix} ] (0, 4τ)

• Page 521. The last displayed equation should read S(N) = T (N) + U(N − 1).

Conditions on k  Conditions on m  N(k, m)
k = 1  m = 1  1
k = 2  m = 0  2
m = 2  1
k > 1 and k ≡ 1 mod 2 m < k \frac{k}{(k,m)}φ((k, m))
k  \frac{1}{2}φ(k)
k ≡ 0 mod 4 m = 0 or m = k φ (\frac{k}{2})
0 < m < k  \frac{2k}{(k,m)}φ (\frac{(k,m)}{2})
k > 2 and k ≡ 2 mod 4 m = 0 φ (\frac{k}{2})
0 < m < k  (2, \frac{m}{2})\frac{k}{(k,m)}φ (\frac{(k,m)}{2})
m = k  \frac{1}{2}φ (\frac{k}{2})

Table 1: FORMULAE FOR N(k, m) IN TERMS OF THE EULER φ-FUNCTION. The integers k and m are assumed to have the same parity for the entries in the table.