We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer $n$, $$\sum _{k=0}^n{(-1)}^k\left[\begin{array}{c}2n-k\\ k\end{array}\right]{(q;q^2)}_{n-k}{q}^{\left(\genfrac{}{}{0pt}{}{k+1}{2}\right)}=\sum _{k=-n}^n{(-1)}^k{q}^{k^2},$$ where $$\left[\begin{array}{c}n\\ m\end{array}\right]=\prod _{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}\text{and}{(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k).$$

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\le 1/d$ of the $L$-functions of any degree $d\ge 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

In this paper we investigate Hesse’s elliptic curves $H_{\mu }:U^3+V^3+W^3=3\mu UVW,\mu \in \mathbf{Q}-\left\{1\right\}$, and construct their twists, $H_{\mu ,t}$ over quadratic fields, and $\tilde{H}(\mu ,t),\mu ,t\in \mathbf{Q}$ over the Galois closures of cubic fields. We also show that $H_{\mu }$ is a twist of $\tilde{H}(\mu ,t)$ over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, $R(t;X):=X^3+tX+t,t\in \mathbf{Q}-\{0,-27/4\}$, to parametrize all of quadratic fields and cubic ones. It should be noted that $\tilde{H}(\mu ,t)$ is a twist of $H_{\mu }$ as algebraic curves because it may not always have any rational points...