follows immediately from Theorem \[fourier1\], , and , noting that $b_m(n,1)=0$ when $n<0$, and that $\alpha(r_1,r_2)-\beta(r_1,r_2)=|r_1+r_2|$ for $r_1,r_2\in {\mathbb{R}}$ with $r_1r_2<0$.

We have $\sqrt{p}L(1,\epsilon_p)=h_F \log({\varepsilon}_0)$, where $h_F$ denotes the class number of $F$ and ${\varepsilon}_0>1$ the fundamental unit.

Twisted Borcherds products {#sect:5}
==========================

As an application of Theorem \[fourier2\] of the previous section we obtain a variant of the Borcherds lift for Hilbert modular surfaces (see [@Bo2], [@Br1], [@BB]). It can be viewed as a multiplicative analogue of the Doi-Naganuma lift [@DN] from holomorphic modular forms of weight $k$ for $\Gamma'$ to Hilbert modular forms of weight $k$ for $\Gamma$. Its existence was suggested by Zagier in [@Za3].

Following [@Za3] §7, we define $$\begin{aligned}
\label{defRp}
R_p(t)=\prod_{b\,(p)} (1-e(b/p)t)^{\epsilon_p(b)}.\end{aligned}$$ It is a rational function of $t$ with coefficients in $F$.

\[Rproperties\] The function $R_p(t)\in F(t)$ has the following properties:

1.  $R_p(t)'=R_p(t)^{-1}$.

2.  $R_p(t^{-1})=R_p(t)$.

3.  $\log (R_p(t))= -\sqrt{p}\sum_{n\geq 1}\frac{1}{n}\epsilon_p(n)t^n$.

The first two properties are verified by direct computation. The third property follows from the identity $\sum_{b\,(p)}\epsilon_p(b) e(bn/p)=\sqrt{p}\epsilon_p(n)$.

\[hilbert\] Let $f=\sum_{n\gg -\infty}c(n)q^n\in {\mathbb{Z}}[j]$ be a weakly holomorphic modular form of weight $0$ for $\Gamma'$ with integral Fourier coefficients. Then there exists a symmetric meromorphic Hilbert modular function $\Psi(z,f)$ for $\Gamma$ (of weight $0$, with 