Multiple standard twists of $L$-functions

1. Problem Statement

The paper addresses the extension of the theory of standard twists of L-functions from the one-dimensional case to the multidimensional case.

• Background: In the Selberg class theory, the standard twist $F(s, \alpha)$ of an L-function $F(s)$ is defined as a Dirichlet series with an exponential factor $e(-\alpha n^{1/d_F})$. It is well-established that $F(s, \alpha)$ admits a meromorphic continuation to the complex plane. Its analytic behavior depends on the spectrum of $F$, denoted $\text{Spec}(F)$. If $\alpha \notin \text{Spec}(F)$, the function is entire; if $\alpha \in \text{Spec}(F)$, it has simple poles at specific points $s^*_\ell$.
• The Gap: The authors seek to define and analyze a multiple standard twist involving a set of $N$ L-functions, $\mathcal{F} = \{F_1, \dots, F_N\}$. The goal is to determine the analytic properties (specifically meromorphic continuation and the structure of singularities) of this multidimensional object, defined on $\mathbb{C}^N$.

2. Definitions and Setup

Let $F_\nu \in \mathcal{S}^\sharp$ (extended Selberg class) for $\nu = 1, \dots, N$, with degrees $d_\nu > 0$, conductors $q_\nu$, and internal shifts $\theta_\nu$.
Let $\kappa_1, \dots, \kappa_N > 0$ be parameters satisfying the constraint:
$$\sum_{\nu=1}^N d_\nu \kappa_\nu = 1$$
The multiple standard twist is defined for $s = (s_1, \dots, s_N) \in \mathbb{C}^N$ (initially in the half-space $\Re(s_\nu) > 1$) by the series:
$$\mathbf{F}(s, \alpha) = \sum_{n_1=1}^\infty \dots \sum_{n_N=1}^\infty \frac{a_1(n_1)\dots a_N(n_N)}{n_1^{s_1} \dots n_N^{s_N}} e\left(-\alpha n_1^{\kappa_1} \dots n_N^{\kappa_N}\right)$$
where $e(x) = \exp(2\pi i x)$.

Key Definitions:

• Spectrum of the Multiple Twist:
  $$\text{Spec}(\mathcal{F}) = \left\{ \prod_{\nu=1}^N \left( \frac{n_\nu}{q_\nu^{\kappa_\nu d_\nu}} \right)^{\kappa_\nu} : \prod_{\nu=1}^N a_\nu(n_\nu) \neq 0 \right\}$$
• Polar Hyperplanes: For integers $\ell \ge 0$, the hyperplanes $H^*_\ell \subset \mathbb{C}^N$ are defined by:
  $$\sum_{\nu=1}^N d_\nu s_\nu = \frac{d+1}{2} - \ell - i\theta$$
  where $d = \sum d_\nu$ and $\theta = \sum d_\nu \theta_\nu$.

3. Methodology

The authors employ a sophisticated analytic approach combining Mellin-Barnes integral representations, functional equations, and asymptotic expansions of Gamma functions.

1. Regularization: They introduce a regularized version of the series, $\mathbf{F}_X(s, \alpha)$, using a parameter $X$ to ensure absolute convergence for all $s \in \mathbb{C}^N$.
2. Integral Representation: Using the Mellin-Barnes integral, they express $\mathbf{F}_X(s, \alpha)$ as a contour integral involving the product of the L-functions $F_\nu(s_\nu + \kappa_\nu w)$ and $\Gamma(w)$.
3. Contour Deformation: The integration path is shifted from the right half-plane to the left, crossing the poles of the integrand. This requires applying the functional equations of the L-functions in $\mathcal{S}^\sharp$ to express $F_\nu$ in terms of $F_\nu(1-s)$.
4. Asymptotic Analysis:
   • They utilize Stirling's formula for the Gamma function to estimate the decay of the integrand.
   • They apply Lemma 1 (an asymptotic expansion of $h_F(s)$) and Lemma 5 (an asymptotic expansion for products of Gamma functions) to handle the singularities arising from the functional equations.
   • Vitali Convergence Theorem is used to prove that the limit as $X \to \infty$ exists and defines a holomorphic function on the resulting domain.
5. Linear Independence: To prove that the singularities are non-trivial (i.e., the residues do not vanish), they use Lemma 8, which establishes the linear independence of exponential functions with distinct linear forms.

4. Key Contributions and Results

Theorem 1: Meromorphic Continuation and Pole Structure

This is the central result describing the global analytic behavior of $\mathbf{F}(s, \alpha)$.

• Case 1 ($\alpha \notin \text{Spec}(\mathcal{F})$): The function $\mathbf{F}(s, \alpha)$ admits an analytic continuation to an entire function on $\mathbb{C}^N$.
• Case 2 ($\alpha \in \text{Spec}(\mathcal{F})$): The function admits a meromorphic continuation to $\mathbb{C}^N \setminus \bigcup_{\ell=0}^\infty H^*_\ell$.
  • The singularities are located on the hyperplanes $H^*_\ell$.
  • For every $\ell \ge 0$ and $s \in H^*_\ell$, the limit
    $$\Xi_\ell(s, \alpha) = \lim_{w \to s, w \notin H^*_\ell} \left( \sum_{\nu=1}^N d_\nu(w_\nu - s_\nu) \right) \mathbf{F}(w, \alpha)$$
    exists and is holomorphic on $H^*_\ell$.
  • Corollary: The quotient of $\mathbf{F}(s, \alpha)$ by the Gamma function $\Gamma(\sum d_\nu s_\nu - \frac{d+1}{2} + i\theta)$ is an entire function.

Theorem 2: Explicit Formula for Residues

The authors provide an explicit expression for the limit functions (residues) $\Xi_\ell(s, \alpha)$:
$$\Xi_\ell(s, \alpha) = \frac{1}{\sqrt{2\pi}} e^{-i \sum (\dots)} \prod_{\nu=1}^N \left( \frac{\kappa_\nu q_\nu^{1/d_\nu}}{2\pi} \right)^{d_\nu(1/2-s_\nu)} W_\ell(s) \Xi(s, \alpha)$$

• $W_\ell(s)$: A polynomial in $s$ of degree $2\ell$.
• $\Xi(s, \alpha)$: A finite sum over indices $n$ such that the spectral parameter $\alpha_n = \alpha$.
• Non-vanishing: Crucially, the authors prove that $\Xi_\ell(s, \alpha)$ does not vanish identically on the hyperplane $H^*_\ell$. This confirms that every hyperplane $H^*_\ell$ is indeed a genuine polar set (a "true" singularity), unlike the 1D case where some poles might be removable depending on structural invariants.

5. Significance and Differences from the 1D Case

The paper highlights significant structural differences between the one-dimensional ($N=1$) and multidimensional ($N \ge 2$) theories:

1. Nature of Singularities:
   • 1D: Singularities are isolated points ($s^*_\ell$). A pole exists at $s^*_\ell$ only if the structural invariant $d_F(\ell) \neq 0$.
   • Multidimensional: Singularities are hyperplanes ($H^*_\ell$). The paper proves that every hyperplane $H^*_\ell$ is a polar set for the multiple twist, regardless of the specific choice of parameters $\kappa_\nu$ (provided they satisfy the constraint).
2. Spectrum Dependence:
   • In the 1D case, the spectrum is fixed by the L-function.
   • In the multidimensional case, the spectrum $\text{Spec}(\mathcal{F})$ depends on the choice of the exponents $\kappa_1, \dots, \kappa_N$. There are infinitely many such choices, leading to infinitely many distinct multiple standard twists for the same set of L-functions.
3. Relation to Multiple Dirichlet Series:
   • While the object resembles a multiple Dirichlet series, the authors note that their techniques (based on Selberg class functional equations and specific asymptotic expansions) differ from the standard methods used in the theory of multiple Dirichlet series (e.g., those by Goldfeld, Diaconu, etc.).

Conclusion

Kaczorowski and Perelli successfully generalize the theory of standard twists to the multidimensional setting. They establish that the multiple standard twist is a meromorphic function on $\mathbb{C}^N$ with singularities confined to a countable family of hyperplanes. The most striking result is that, unlike the 1D case where poles can be absent, the multidimensional twist exhibits a "full" set of singularities on these hyperplanes, providing a robust framework for studying the analytic properties of products of L-functions twisted by exponential factors.