Next, we turn to the \(\sigma \)-integral of (4.34). We first explicate our choice of integration contour \(\Gamma _\sigma (D)\) in (2.5) for a given \(n_2 \ge 1\) pole D in \(P \cup \} \}\).

5.1 Contour of Integration

The contour \(\Gamma _\sigma (D)\) defined in the \(\sigma \) upper half plane restricted to \(\sigma _1 \in [0,1)\) crosses locus (3.16) associated with quadratic poles. Writing out the real and imaginary parts of (3.16) gives

$$\begin & -ac\gamma (\rho _1\sigma _1-v_1^2-\rho _2\sigma _2+v_2^2)\nonumber \\ & \quad +(ad+bc)v_1+(-bd\alpha -\gamma \Sigma )\sigma _1 -ac\delta \rho _1-bd\beta -\delta \Sigma =0, \nonumber \\ & \quad -ac\gamma (\rho _1\sigma _2+\rho _2\sigma _1-2v_1v_2)+(ad+bc)v_2 \nonumber \\ & \quad +(-bd\alpha -\gamma \Sigma )\sigma _2-ac\delta \rho _2= 0. \end$$

(5.1)

Solving the second equation for \(\rho _1\) yields

$$\begin \rho _1= \frac. \qquad \end$$

(5.2)

Now we recall that \(v_1\) lies in range (4.3), which we write as

$$\begin v_1 = \frac(\gamma \sigma _1+\delta )-\frac+x, \end$$

(5.3)

where \(x\in (0,-1/ac\gamma )\). Inserting (5.3) as well as (5.2) into the first equation of (5.1) gives

$$\begin \left( \rho _2 \sigma _2 - v_2^2 \right) \left( \sigma _1 + \frac \right) ^2 + \sigma _2^2 \left( x + \frac \right) ^2 = \sigma _2^2 \left[ \frac - \left( \rho _2 \sigma _2 - v_2^2 \right) \right] . \nonumber \\ \end$$

(5.4)

This describes an ellipse in the \((\sigma _1, x)\)-plane provided that the right-hand side of this equation is non-vanishing and positive,

$$\begin \rho _2\sigma _2-v_2^2 = \rho _2\sigma _2 - \left( \frac \right) ^2 \sigma _2^2 <\frac, \end$$

(5.5)

where we set

$$\begin \frac = - \frac. \end$$

(5.6)

Combining the \(\sigma _2\)-direction with (5.4) results in an ellipsoid in three dimensions, with \(\sigma _2\) taking values in the range specified by (5.5).

On the other hand, we note that the point \((\sigma _1 = - \delta /\gamma , x, \sigma _2 =0)\) lies in locus (5.4). This point, which lies on the boundary of the Siegel upper half plane, is the anchoring point of a curve in the complex \(\sigma \)-plane at fixed \(x\in (0,-1/ac\gamma )\), as follows. Since \(\sigma _2 >0\), we may divide (5.4) to arrive at

$$\begin \left( \sigma _1 + \, \frac \right) ^2 + \left( \sigma _2 - \frac \right) ^2 = \frac, \end$$

(5.7)

where we defined

$$\begin \lambda \equiv \rho _2 - \left( \frac \right) ^2 \sigma _2 > 0 \;\;,\;\;\; X(x) \equiv \frac - \left( x + \frac \right) ^2. \end$$

(5.8)

Note that \(X(x)> 0\) for \(x\in (0,-1/ac\gamma )\), and that the positivity of \(\lambda \) can be enforced by taking \(\rho _2\) to be sufficiently large. At fixed x, above equation (5.7) describes a circle in the complex \(\sigma \)-plane anchored at \((\sigma _1 = - \delta /\gamma , \sigma _2 =0)\), provided that \(\lambda \) is kept fixed. The latter is compatible with condition (5.5), as follows. We write (5.5) as

$$\begin 0< \sigma _2 < \frac. \end$$

(5.9)

The bound on the right-hand side is precisely saturated when \(\sigma _2 = X_ /\lambda \), where \(X_ \) is the maximal value of X, which is attained for \(x = - 1/(2 a c \gamma )\). The associated point on the circle is \((\sigma _1 = - \delta /\gamma , \sigma _2 = X_ /\lambda )\), which is the point on the circle that intersects the line \(\sigma _1 = - \delta /\gamma \) in the complex \(\sigma \)-plane. Thus, we see that keeping \(\lambda \) fixed is compatible with restricting the range of \(\sigma _2\) to

$$\begin 0 < \sigma _2 \le \frac. \end$$

(5.10)

At fixed \(\lambda \), the circle described by (5.7) is homotopic to a Ford circle \(}( -\delta , \gamma )\) in the complex \(\sigma \)-plane anchored on the real axis at \(\sigma _1 = - \delta /\gamma \) (see Appendix C.1 for details). Note that since the homotopy is between circles, the leading behaviour of the integrand in (5.11) when approaching the point \((\sigma _1 = - \delta /\gamma , \sigma _2 =0)\) along any of these two circles in the same. The chosen range \(\sigma _1 \in [0,1)\) constrains the poles contributing to (4.34) to those associated with \(0 \le -\frac< 1\). Since this holds for any \(x\in (0,-1/ac\gamma )\), our integration contour over \(\sigma \) for a given pole is \(\Gamma _\sigma (D) = }( -\delta , \gamma ) \), which for notational simplicity we will denote by \(\Gamma _\).

The interpretation of this construction is the one given in [21]. When \(\sigma _2\) is large, the integration contour does not intersect the ellipsoid described above. When lowering the value of \(\sigma _2\), the integration contour will cross some of the poles in the Siegel upper half plane described by (5.7). This will cease to be the case when \(\sigma _2\) reaches the boundary \(\sigma _2 = 0\) of the Siegel upper half plane. Fixing the value of \(\lambda \) to be large enough, we note that as we decrease \(\sigma _2\) we continue to remain in the \(}}\)-chamber, ensuring that the integration contour does not cross any \(n_2=0\) pole.

We will now perform the \(\sigma \)-integral of (4.34) over the Ford circle \(\Gamma _\) described above, following the prescription given in (2.5), which results in

$$\begin&(-1)^ \sum _ P\\ b \in }/a} \end} \int \limits _} \textrm \sigma \; } \sum _ M,N\ge -1\\ L\in } \end} \, L \, d(M) \, d(N) \nonumber \\&\quad e^\frac}}+ \frac\frac \right) } \, \psi (\Gamma )_}\ell } \; e^}}\frac\frac +\frac}\frac \right] \right) } \nonumber \\&\quad \frac \sqrt \left( \text \left[ \sqrt}\left( -\frac-\frac-\frac}} \right) \right] \right. \nonumber \\&\quad \left. -\text \left[ \sqrt}\left( -\frac-\frac}} \right) \right] \right) . \end$$

(5.11)

To perform this integral, we will use the decomposition of the error functions given above.

5.2 Bessel Function \(I_\)

We first focus on the constant terms in decompositions (4.42) and (4.43).

Firstly we show that the two cases corresponding to \(X=0\) and \(Y=0\) give rise to the same contribution, as follows. The condition \(X=0\) yields \(m -a^2\,M = - c^2 N\), and hence \(L = 2 c N / a\). We will show later that only terms with \(}}} <0\) contribute. Therefore combining \(L = 2 c N / a\) with \(}}} <0\) results in \(L > 0\), which in turn implies \(N=-1\). The latter implies a|2. Consequently \(}}} = - b \, 2\,m /a = k m, k \in }\). Therefore \(}}} = - }}} \, \text \, 2\,m \) and hence \( \psi (\Gamma )_}\ell } = \psi (\Gamma )_}\ell } \). Using mapping (4.48), this shows that the contribution from the sector \(X=0\) equals the one from the sector \(Y=0\). Therefore, the combined contribution from the sectors \(X Y \ge 0\) can be expressed as follows,

$$\begin \begin&(-1)^ \sum _ P\\ b \in }/a} \end} \int \limits _} d \sigma \; } \sum _ M,N\ge -1\\ L\in } \\ 0 \le \frac+\frac}} < -\frac \end} L \, d(M) \, d(N) \\&\quad e^\frac}}+ \frac\frac \right) } \, \psi (\Gamma )_}\ell } \; \, e^}}\frac\frac +\frac}\frac \right] \right) } , \end \end$$

(5.12)

where continued fraction condition (4.44) now takes the form given in (5.12). To proceed, we interchange the integration with the summation over M, N. This is allowed by the following arguments.

First we note that the condition \(0 \le \frac+\frac}} < -\frac\) in the summation above can be written as

$$\begin -m < a^2 M -c^2N \le m. \end$$

(5.13)

Then, using the expression for L given in (4.13), we write out \(} = 4MN-L^2\) and obtain

$$\begin } = \frac \left[ -(a^2M-c^2N)^2-\left( m^2-2m(a^2M+c^2N) \right) \right] . \end$$

(5.14)

Now let us consider terms that satisfy \(}\le 0\), in which case we obtain from (5.14),

$$\begin 2m(a^2M+c^2N) \le (a^2M-c^2N)^2+m^2 . \end$$

(5.15)

Combining this with (5.13) we infer

$$\begin a^2M+c^2N \le m. \end$$

(5.16)

Then, by combining this last inequality with (5.13) we obtain the bounds

$$\begin a^2 M \le m, \hspace c^2 N < m . \end$$

(5.17)

Therefore, for a given a, c there is only a finite set of values M, N which satisfy \(} \le 0 \) as well as continued fraction condition (4.44) and the condition \(L \in }\). Thus, in this case, we can interchange the integration with the summation over M, N.

Next, let us consider the terms with \(} >0 \). For large values of M, the Fourier coefficients of \(1/\eta ^\) grow exponentially as

$$\begin d(M) \sim e^}. \end$$

(5.18)

From (5.14), and using (5.13), we infer that for large M, N, \(} \) behaves, schematically, as \(} \sim M + N\), and hence becomes large. Parametrizing the Ford circle \(\Gamma _\) in (5.12) by

$$\begin \sigma (\theta ) = -\frac+\frac\left( \frac} \right) \end$$

(5.19)

or, equivalently, by

$$\begin \sigma _1(\theta ) = -\frac-\frac\frac, \hspace \sigma _2(\theta ) = \frac\frac, \end$$

(5.20)

where \(\theta \in [0,\pi ) \cup (\pi , 2 \pi )\), we infer that on the contour \(\Gamma _\),

$$\begin \Big \vert e^}}\frac\frac +\frac}\frac \right] } \Big \vert \le e^ }}} + \frac \Delta }. \end$$

(5.21)

Since \(d(M) \, d(N) \, e^ }}}} \) is exponentially suppressed for large M, N, the sum over M, N in (5.12) is uniformly convergent on \(\Gamma \) by the Weierstrass M test, and since each summand is integrable, we conclude that interchanging the integration with the summation over M, N is justified also when \(} >0 \).

Thus, interchanging the integration with the summation over M, N results in

$$\begin \begin&(-1)^ \sum _ P\\ b \in }/a} \end} \;\; \sum _ M,N\ge -1 \\ -m < a^2M - c^2N \le m \\ L\in } \end} L \, d(M) \, d(N) \, e^\frac}}+ \frac\frac \right) } \,\psi (\Gamma )_}\ell } \\&\quad \int \limits _} \textrm \sigma \; } \, e^}}\frac\frac +\frac}\frac \right) } . \end \end$$

(5.22)

5.2.1 Bessel Integral

We perform the \(\sigma \)-integration over the Ford circle \(\Gamma _\) that skirts the point \(- \delta /\gamma \). This Ford circle has radius \(1/(2 \gamma ^2)\), is centred at \(\sigma = - \frac + i \frac\),

$$\begin \Gamma _: \vert \sigma + \frac - i \frac \vert = \frac , \end$$

(5.23)

and is oriented counter clockwise. Then, (5.22) becomes replaced by

$$\begin \begin&(-1)^ \sum _ P' \\ b \in }/a } \end} \;\; \sum _ M,N\ge -1 \\ -m < a^2M - c^2N \le m \\ L\in } \end} L \, d(M) \, d(N) \, e^\frac}}+ \frac\frac \right) } \, \psi (\Gamma )_}\ell }\\&\quad \int \limits _} \textrm \sigma } \, e^}}\frac\frac +\frac}\frac \right) } , \end \end$$

(5.24)

where \(P'\) denotes the set P, but with \(\delta \) restricted to lie in the range \(0 \le -\delta < \gamma \). We change the integration variable to

$$\begin }}} = \gamma \left( \gamma \sigma + \delta \right) . \end$$

(5.25)

The essential singularity is now located at the origin \(}}} = 0\). We choose the branch cut, which originates at \(}}} = 0\), to lie along the negative imaginary axis of the \(}\)-plane. Next, we change the integration variable once more,

$$\begin w = \frac \frac}}} . \end$$

(5.26)

Now the branch cut originates at \(w = 0\) and lies along the negative real axis of the w-plane. The integration contour now runs along a line parallel to the imaginary axis,

$$\begin - \frac}} \int \limits _}}}-i\infty }^}}}+i \infty } \frac} \; e^} \Delta } w +\frac} \right) } \end$$

(5.27)

with \(}}} >0\). Now recall that \(\Delta >0\). When \(}} } \ge 0\), the coefficient \(} \Delta } / (4m \gamma )\) in the exponent is \(\ge 0\), and hence the integration contour can be closed in the half plane \(\textrm \, w > 0\), where the integrand is analytic and hence the integral vanishes. Thus, we now take \(}}} < 0\).

Then, performing the redefinition

$$\begin t = \frac}}}| \Delta } \, w \; \end$$

(5.28)

and defining \(z = \frac\sqrt}}|\Delta }\), we obtain for integral (5.27), with \(\epsilon > 0\),

$$\begin & - \frac} \left( \frac}}}| } \right) ^ \ \int \limits _^ \frac t }} \; e^ }\nonumber \\ & \qquad = - 2 \pi \, \frac} \left( \frac}}} |} \right) ^ \, I_ \left( \frac \, \sqrt}}}| \Delta } \right) , \end$$

(5.29)

where \(I_\nu (z)\) denotes the modified Bessel function of first kind of index \(\nu \),

$$\begin I_\nu (z) = \fracz)^\nu }\int \limits _^ \textrm t \,\,t^ e^, \end$$

(5.30)

where \(\epsilon > 0\). Then, (5.22) becomes

$$\begin & (-1)^ \, i^ \, 2 \pi \sum _ P' \\ b \in }/a } \end} \;\;\; \sum _ M,N\ge -1 \\ -m< a^2M - c^2N \le m \\ L\in }, \; }}} < 0 \end}\nonumber \\ & \quad \psi (\Gamma )_}\ell } \; L \, d(M) \, d(N) \, \frac\frac}}+ \frac\frac \right) }} \left( \frac}}} |} \right) ^ \,\nonumber \\ & \quad I_ \left( \frac \, \sqrt}}}| \Delta } \right) . \end$$

(5.31)

Note that in (5.31) the dependence on \(\alpha \) and \(\delta \) is entirely encoded in multiplier system \(\psi (\Gamma )_}\ell }\) and in the phase

$$\begin e^\frac}}+ \frac\frac \right) }. \end$$

(5.32)

Since \(0\le - \delta < \gamma \) and \(\alpha \in }/\gamma }\), and since \(\alpha \) is the modular inverse of \(\delta \), i.e. \(\alpha \delta = 1 \text \gamma \), each \(\delta \) uniquely specifies one \(\alpha \). Thus, the sum over \(\delta \) yields the generalized Kloosterman sum \(\textrm( \frac, \frac}};\gamma ,\psi )_}}\),

$$\begin \textrm \left( \frac, \frac}};\gamma ,\psi \right) _}} = \sum _ 0\le -\delta <\gamma \\ (\delta ,\gamma )=1, \alpha \delta = 1 \text \gamma \end}e^\frac}} +\frac\frac\right) }\psi (\Gamma )_}\ell }. \qquad \end$$

(5.33)

Thus, (5.31) can be written as

$$\begin&(-1)^ i^ \, 2 \pi \sum _^ \; \sum _ S_G \\ b \in }/a } \end} \;\; \sum _ M, N \ge -1 \\ -m< a^2M - c^2N \le m \\ L \in }, \, }<0 \end} L \, d(M)d(N) \; \frac(\frac \frac}},\gamma ,\psi )_}}} \nonumber \\&\quad \left( \frac} \vert } \right) ^ I_\left( \frac\sqrt}\vert } \right) , \end$$

(5.34)

where \(} = 4 M N - L^2\), with L given in (4.13). Note that the sum over the allowed M, N is finite.

Next, using (4.17), we express the triplet (N, L, M) in terms of the triplet \((}}}, m, }}})\). We then trade the sum over M, N for a sum over \(}}}, }}}\). In Sect. 5.3 we will show that \(}}}\) is bounded by \(}}} \ge -1\). Writing \(}}}\) as \(} = 4\,m }}} - \ell ^2\), we rewrite (5.34) as

$$\begin&(-1)^ i^ \, 2 \pi \sum _^ \; \sum _ }}} \ge -1 \\ }}} \in }/2m } \\ }}}< 0 \end}\;\; \sum _ a>0, c<0 \\ b \in }/ a }, \; a d - b c = 1 \\ 0 \le \frac + \frac}}} < - \frac \end} \nonumber \\&\qquad \left( (ad + bc ) }}} + 2 ac }}} + 2 bd m \right) d( c^2 }}} + d^2 m + cd }}}) \, d(a^2 }}} + b^2 m + ab }}}) \nonumber \\&\qquad \frac(\frac \frac}},\gamma ,\psi )_}}} \left( \frac} \vert } \right) ^ I_\left( \frac\sqrt}\vert } \right) , \end$$

(5.35)

which we write as

$$\begin (-1)^ i^ \, 2 \pi \sum _^ \; \sum _ }}} \ge -1 \\ }}} \in }/2m } \\ }}} < 0 \end} \;\; c_m^F(},}) \, \frac(\frac \, \frac}},\gamma ,\psi )_}}} \left( \frac} \vert } \right) ^ I_\left( \frac\sqrt}\vert } \right) ,\nonumber \\ \end$$

(5.36)

where \(c_m^F(},}) \) is defined by

$$\begin c_m^F(},})= & \sum _ a>0, c<0 \\ b \in }/ a }, \; a d - b c = 1 \\ 0 \le \frac + \frac}}} < - \frac \end} \left( (ad + bc ) }}} + 2 ac }}} + 2 bd m \right) \nonumber \\ & \, d( c^2 }}} + d^2 m + cd }}}) \, d(a^2 }}} + b^2 m + ab }}}). \end$$

(5.37)

Note that the above sum includes two subsets of matrices in \(S_G\). The first subset contains matrices satisfying \(}/2\,m=-b/a\), while the second subset contains matrices that correspond to the continued fraction expansion of \(}/2\,m\). The latter subset is finite by definition, while the bounds \(M,N \ge -1\) and (5.13) can be used to show the finiteness of the first subset. This is consistent with the proofs of finiteness of [9, 10].

5.3 Lower Bound on \(}\)

We next show that \(} \ge -1\) whenever the condition

$$\begin 0 \le \frac + \frac}}} < - \frac \end$$

(5.38)

is true. To this end, we first recall that (5.38) can be written as

$$\begin -m < a^2 M - c^2 N \le m. \end$$

(5.39)

and further \(M, N \ge -1\). Then beginning with

$$\begin }}} = \frac m - \frac M + \frac N, \end$$

(5.40)

we study three cases:

1.

\(d=0\): Then, (5.40) becomes

$$\begin }}} = - \frac M = \frac, \end$$

(5.41)

which satisfies \(}}} \ge -1\), since \(M \ge -1\) and \(c^2 \ge 1\).

2.

\(bc<0\): In this case, we obtain,

$$\begin (ac)^2 }}}= & a b c d m - b c a^2 M + d a c^2 N \nonumber \\= & a b c d m + c^2 N - b c (a^2 M - c^2 N ) \nonumber \\= & a b c d m + c^2 N + | b c | (a^2 M - c^2 N ). \end$$

(5.42)

Using the lower bound in (5.39), we get,

$$\begin (ac)^2 }}} > a b c d m + c^2 N - | b c | m= & c^2 N + m b c \left( a d + 1 \right) \nonumber \\= & c^2 N + m \left( (a d)^2 - 1 \right) \ge c^2 N, \end$$

(5.43)

Since \((ad)^2 \ge 1\),

$$\begin } > \frac \ge -1. \end$$

(5.44)

3.

\(bc \ge 0\): Here we have, using the upper bound in (5.39),

$$\begin (ac)^2 }}}= & a b c d m - b c a^2 M + d a c^2 N \nonumber \\= & a b c d m + c^2 N - b c (a^2 M - c^2 N ) \nonumber \\\ge & a b c d m + c^2 N - b c m \nonumber \\= & c^2 N + b c m (a d -1) = c^2 N + (b c)^2 m \ge c^2 N. \end$$

(5.45)

Hence,

$$\begin } \ge \frac \ge -1. \end$$

(5.46)

5.4 Bessel Function \(I_\)

Now we focus on the terms \(E_\) in decompositions (4.42), (4.43) and (4.46). In Sect. 5.5, we will show that these terms only give a non-vanishing contribution when \(X Y \ne 0\), whereas when \(X Y =0\) they cancel out. Thus, we will assume \(X Y \ne 0\) in the following.

We proceed as in Sect. 5.2.1, namely, we restrict the sum over \(\delta \) to the range \(0 \le -\delta < \gamma \) and perform the integration over the Ford circle \(\Gamma _\) in (5.23). Collecting the terms proportional \(E_\) in (5.11), and changing the integration variable to \(}}}\) given in (5.25), we obtain

$$\begin \begin&(-1)^\sum _} \, \textrm ( \frac, \frac}};\gamma ,\psi )_}} \; L \, d(M) \, d(N) \; \int \limits _}}} \textrm} \;\frac}^}}}\, e^}}\frac}} +\frac}\frac}} \right] \right) } \\&\quad \frac\left( E_}}\left( \frac+\frac}}\right) +E_}}\left( -\frac-\frac-\frac}}\right) \right) , \end\nonumber \\ \end$$

(5.47)

where L and \(}}}\) are given in (4.13), and where the generalized Kloosterman sum \(\textrm ( \frac, \frac}};\gamma ,\psi )_}}\) is given in (5.33). The set \(P_0\) is given by

$$\begin _0 = \left\ a & b \\ c&d \end \in \textrm(2,}), M, N, \gamma \in } \; \vert a,\gamma >0, c<0, M \ge -1, N \ge -1, L \in }, X Y \ne 0 \right\} .\nonumber \\ \end$$

(5.48)

The integration contour \(}}}\) denotes a Ford circle centred at \(}}} = i/2\) that skirts the origin \(} =0\).

Using symmetry property (4.54), we write (5.47) as

$$\begin \begin&(-1)^\sum _} \, \left[ \textrm ( \frac, \frac}};\gamma ,\psi )_}} + \textrm ( \frac, \frac}};\gamma ,\psi )_}) } \right] \; L \, d(M) \, d(N) \; \\&\quad \int \limits _}}} \textrm} \;\frac}^}}}\, e^}}\frac}} +\frac}\frac}} \right] \right) } \; \frac E_}}\left( \frac+\frac}}\right) . \end \end$$

(5.49)

Using the expression for \(E_}}\) given in (4.38) and the relation

$$\begin \frac}}+m\left( \frac+\frac}}\right) ^2 = \frac , \end$$

(5.50)

we obtain

$$\begin \begin&(-1)^\sum _} \, \left[ \textrm ( \frac, \frac}};\gamma ,\psi )_}} + \textrm ( \frac, \frac}};\gamma ,\psi )_}) } \right] \, \frac+\frac}} \right) } \, d(M) \, d(N) \\&\quad \int \limits _}}} \textrm} \;\frac}^}}}\frac}\, e^} \frac}} + \frac \frac}} \right) }. \end \end$$

(5.51)

Note that the integrand does not exhibit a branch cut. Performing the variable change given in (5.26), the integral over the Ford circle \(}}\) takes a form similar to (5.27), with \(}}}/4m\) replaced by \(N/a^2\). The integral will be non-vanishing provided \(N/a^2 <0\). This in turn implies \(N=-1\). As shown in Sect. 5.5, the only contributions to the sum come from the terms in set the \(P_0\) satisfying \(a=1\) and \(M=m\). Using the expression for L and \(}}}\) given in (4.13), we infer that in this case,

$$\begin \frac+\frac}} = \frac \;\;,\;\;\; L = -c>0, \end$$

(5.52)

in which case

$$\begin \frac+\frac}}} = -2m, \end$$

(5.53)

so that (5.51) yields (using \(d(-1) = 1\))

$$\begin \begin&(-1)^\sum _'} \left[ \textrm ( \frac, \frac}};\gamma ,\psi )_}} + \textrm ( \frac, \frac}};\gamma ,\psi )_}) } \right] \; d(m) \\&\quad \int \limits _}}} \textrm} \;\frac}^}}}\sqrt}\, e^}\frac}} - \frac}} \right) }, \end \end$$

(5.54)

where the set \(P_'\) is the set \(P_0\) subject to the restrictions \(a=1,N=-1,M=m\).

From (5.52) we infer

$$\begin } = 4MN-L^2 = -4m-c^2 , \end$$

(5.55)

in which case

$$\begin \frac}} = -1-\frac. \end$$

(5.56)

Using (5.52), we can write the Kloosterman sums in (5.54) as (recall that \(b \in }/a} \) with \(a=1\) fixes b to a single value)

$$\begin (-1)^ \sum _ c<0 \end} \;\; \sum _ 0\le -\delta <\gamma \\ (\delta ,\gamma )=1, \alpha \delta = 1 \text \gamma \end}e^\frac}} +\frac\frac\right) } \left( \psi (\Gamma )_} + \psi (\Gamma )_} \right) . \nonumber \\ \end$$

(5.57)

Using

$$\begin \sum _ c<0 \end} f(c ) = \sum _}/2m}} \;\; \sum _ c < 0 \\ c = j \; \text \; 2m \end} \, f(c) . \end$$

(5.58)

and using the multiplier system property \(\psi (\Gamma )_} = \psi (\Gamma )_}\) (with \(k \in }\)) [20], we write (5.57) as

$$\begin (-1)^ \sum _}/2m}}\sum _ c<0\\ c = j \text 2m \end}\sum _ 0\le -\delta <\gamma \\ (\delta ,\gamma )=1, \alpha \delta = 1 \text \gamma \end}e^\frac}} +\frac\frac\right) } \left( \psi (\Gamma )_}+ \psi (\Gamma )_} \right) . \nonumber \\ \end$$

(5.59)

Using (5.56), this becomes

$$\begin & (-1)^ \sum _}/2m}}\sum _ c<0\\ c = j \text 2m \end}\sum _ 0\le -\delta <\gamma \\ (\delta ,\gamma )=1, \alpha \delta = 1 \text \gamma \end} e^\frac} \, e^ +\frac\frac\right) }\nonumber \\ & \qquad \left( \psi (\Gamma )_}+ \psi (\Gamma )_} \right) . \end$$

(5.60)

Next we focus on the sum over c in this expression,

$$\begin \sum _}/2m}}\sum _ c<0\\ c = j \text 2m \end} e^\frac} \left( \psi (\Gamma )_} + \psi (\Gamma )_}\right) , \end$$

(5.61)

which we write as

$$\begin \frac \sum _}/2m}}\sum _ c \in } \backslash \ \\ c = j \text 2m \end} e^\frac} \left( \psi (\Gamma )_} + \psi (\Gamma )_} \right) , \end$$

(5.62)

where we made use of the symmetry \(j \rightarrow - j\) to obtain an expression with the symmetry \(c \rightarrow -c\).

Expression (5.62) is divergent. Below we will discuss a regularization procedure to extract a finite part of this expression.

5.4.1 Regularization

We now regard the sum over \(c \in } \backslash \ \) in (5.62),

$$\begin \sum _ c \in } \backslash \ \\ c = j \text 2m \end} e^\frac}, \end$$

(5.63)

as

$$\begin \lim _ \left( \vartheta _(\tau )-\delta _ \right) , \end$$

(5.64)

where \( \vartheta _(\tau ) \) denotes the standard weight 1/2 index m Jacobi theta function. Using this, we then regard (5.62) as

$$\begin \frac \lim _ \sum _}/2m}} \left( \vartheta _(\tau )-\delta _ \right) \left( \psi (\Gamma )_} + \psi (\Gamma )_} \right) . \end$$

(5.65)

We focus on the combination

$$\begin \sum _}/2m}} \left( \vartheta _(\tau )-\delta _ \right) \psi (\Gamma )_}, \end$$

(5.66)

which is one of the combinations contained in (5.65). Using that \(\psi (\Gamma )_}\) are the components of a unitary matrix, i.e.

$$\begin \psi (\Gamma ) = \overline) \right) ^T}, \end$$

(5.67)

we infer the property

$$\begin \psi (\Gamma )_} = \psi (}}})_ j}, \end$$

(5.68)

where

$$\begin \Gamma = \begin \alpha & \beta \\ \gamma & \delta \end \quad , \quad }}} = \begin \delta & \beta \\ \gamma & \alpha \end \,. \end$$

(5.69)

Now we recall the transformation property under \(\textrm(2,})\) transformations,

$$\begin \vartheta _\left( \frac, \frac \right) = (\gamma \tau +\delta )^e^} \sum _}/2m}} \psi (\Gamma )_\;\vartheta _(\tau ,v).\nonumber \\ \end$$

(5.70)

Setting \(v=0\) and choosing the \(\textrm(2,})\) transformation \(}}}\), this becomes

$$\begin \vartheta _\left( \frac \right) = (\gamma \tau +\alpha )^ \sum _}/2m}} \psi (}}})_\;\vartheta _(\tau ). \end$$

(5.71)

Then, by combining (5.66) with (5.68) and (5.71), we obtain

$$\begin \sum _}/2m}} \left( \vartheta _(\tau )-\delta _ \right) \psi (\Gamma )_} = (\gamma \tau +\alpha )^ \, \vartheta _\left( \frac \right) - \psi (\Gamma )_}.\qquad \end$$

(5.72)

Now we study this equation in the limit \(\tau \rightarrow -\alpha /\gamma \). Using

$$\begin \lim _ \vartheta _(\tau ) = 0 \;\; \text \ell \ne 0 \mod 2m, \\ 1 \;\; \text \ell = 0 \mod 2m, \\ \end\right. } \end$$

(5.73)

we infer that in the limit \(\tau \rightarrow -\alpha /\gamma \), the right-hand side of (5.72) tends to \(- \psi (\Gamma )_}\) when \(\ell \ne 0\), whereas when \(\ell = 0\) it diverges and behaves as \((\gamma \tau +\alpha )^ - \psi (\Gamma )_\). We note that the divergent term \((\gamma \tau +\alpha )^\) that arises when \(\ell =0\), is due to the presence of a constant term in \(\vartheta _(\tau )\). Then, by subtracting this constant term we obtain the following regularized expression,

$$\begin \left( \lim _ \sum _}/2m}} \left( \vartheta _(\tau )-\delta _ \right) \psi (\Gamma )_} \right) \vert _} = - \psi (\Gamma )_}. \end$$

(5.74)

A similar reasoning applies to the other combination, proportional to \(\psi (\Gamma )_}\), contained in (5.65).

Thus, we are led to the following regularized expression for (5.65),

$$\begin \frac \lim _ \sum _}/2m}} \left( \vartheta _(\tau )-\delta _ \right) \left( \psi (\Gamma )_} + \psi (\Gamma )_} \right) = - \psi (\Gamma )_}.\nonumber \\ \end$$

(5.75)

Using this in (5.60), we arrive at our proposal for the regularized expression for (5.60),

$$\begin (-1)^ \sum _ 0\le -\delta <\gamma \\ (\delta ,\gamma )=1, \alpha \delta = 1 \text \gamma \end}e^ +\frac\frac\right) } \psi (\Gamma )_} = (-1)^ \, \textrm ( \frac, -1;\gamma ,\psi )_ . \nonumber \\ \end$$

(5.76)

To summarize, the regularization procedure described above removes one divergent contribution that arises when \(\ell =0\). At present we do not have neither a physics nor a mathematics justification for using precisely this regulator.

5.4.2 Bessel Integral

Now we return to (5.54), which we regularize using expression (5.76),

$$\begin \begin (-1)^ \; d(m) \, \sqrt} \, \sum _^ \textrm ( \frac, -1;\gamma ,\psi )_ \, \int \limits _}}} d} \;\frac}^}}} \, e^}\frac}} - \frac}} \right) }. \end\nonumber \\ \end$$

(5.77)

Changing the integration variable to

$$\begin t = 2\pi \frac}}, \end$$

(5.78)

we get

$$\begin \begin (-1)^ \; d(m) \, \sqrt} \, \frac}} \, \sum _^ \textrm ( \frac, -1;\gamma ,\psi )_ \, \int \limits _^ \textrm t \frac} \, e^\left( \frac}} \right) ^2+t} . \end\nonumber \\ \end$$

(5.79)

Using (5.30) we obtain,

$$\begin (-1)^\sqrt\,i^\, d(m) \sum _ \frac ( \frac, -1;\gamma ,\psi )_}}\left( \frac \right) ^6 I_\left( \frac\sqrt} \right) .\nonumber \\ \end$$

(5.80)

5.5 Isolating Non-vanishing Contributions

We return to (4.54) show that the sum exhibits cancellations between various terms, thereby identifying non-vanishing contributions. Taking into account the form of \( E_ \left( X\right) \) and \(I_\left( X \right) \) given in (4.38) and (4.39), we write (4.54) as

$$\begin \begin&(-1)^ \sum _ P\\ b \in }/a } \end} \; } \sum _ M,N\ge -1\\ L\in }, \; X \ne 0 \end} \frac \, d(M) \, d(N) \\&\quad e^\frac}}+ \frac\frac \right) }\left( \psi (\Gamma )_}\ell }+\psi (\Gamma )_}\ell }\right) e^}}\frac\frac +\frac}\frac \right] \right) } F_}(X^2) , \end \nonumber \\ \end$$

(5.81)

where \(F_\) is a function that only depends on