A generalization of the Mordell integral

TL;DR

The paper generalizes the Mordell integral by introducing $h_N(z;\tau,\chi)=\int_{-\infty}^{\infty}\Phi_N(x;\chi)\,e^{\pi i\tau x^2-2\pi zx}\,dx$, where $\Phi_N$ encodes a Dirichlet character twist. It establishes a comprehensive set of analytic and transformation properties for this generalized integral, including holomorphy in $z$, parity determined by $\chi(-1)$, shift relations in $z$ with explicit Gauss-sum coefficients, and a modular-type transformation under $\tau\mapsto -1/\tau$ that relates $h_N$ with conjugate characters. The work builds the necessary Fourier-analytic framework via the kernel $\Phi_N$ and Gauss sums, recovering special cases (e.g., $N=4$ yields a multiple of the classical Mordell integral) and enabling connections to the theory of mock modular forms as influenced by Zwegers. It provides a foundation for further study of $\chi$-twisted Mordell-type integrals and their applications in analytic number theory.

Abstract

We find a generalization of the Mordell integral and we also establish a set of properties for a generalization of the Mordell integral similar to those in the third author's PhD thesis.

Theorems & Definitions (18)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5
• proof