Equivalence between the Functional Equation and Voronoï-type summation identities for a class of $L$-Functions

Abstract: To date, the best methods for estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with nice" test functions $f(n)$, provided $a(n)$ is anarithmetic function". These arithmetic functions $a(n)$ are called so because they are expected to appear as coefficients of some $L$-functions satisfying certain properties. It has been well-known that the functional equation for a general $L$-function can be used to derive a Voronoï-type summation identity for that $L$-function. In this article, we show that such a Voronoï-type summation identity in fact endows the $L$-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

• The paper establishes that a Voronoï-type summation for coefficients forces the Dirichlet series to satisfy a Hecke-type functional equation.
• It introduces an auxiliary modular relation via Mellin transforms that links analytic properties with discrete arithmetic coefficients.
• The result unifies various summation identities and forms a framework applicable to both automorphic and non-automorphic L-functions.

Equivalence Between Functional Equation and Voronoï-Type Summation for General $L$-Functions

Introduction and Motivation

The interrelation between the analytic properties of $L$-functions and their arithmetic coefficients underpins much of analytic number theory. The classical Voronoï summation formula, which provides deep links between the analytic continuation/functional equation of Dirichlet series and "discrete-to-continuous" transforms of their coefficients, has driven significant progress in bounding problems such as the Dirichlet divisor problem and the Gauss circle problem. Traditionally, the functional equation for an $L$-function with suitable gamma factors leads to a Voronoï-type summation formula for its coefficients. However, the converse direction—whether a Voronoï-type summation formula implies the functional equation—remained far less explored for broad classes of $L$-functions.

This paper establishes a precise equivalence between the functional equation of Hecke type, generalized Voronoï-type summation identities (notably Riesz-type formulas), and modular-type relations for a large class of Dirichlet series incorporating multiple gamma factors. The main contribution is to show that the presence of a Voronoï-type summation for the coefficients unconditionally forces the underlying Dirichlet series to satisfy a corresponding analytic functional equation, thus offering a new structural characterization of $L$-functions in this context.

Framework, Definitions, and Main Results

The Class of Dirichlet Series

Define $\mathcal{C}$ as the class of Dirichlet series $\phi(s)=\sum_{n=1}^\infty a_n\lambda_n^{-s}$ and $\psi(s)=\sum_{n=1}^\infty b_n\mu_n^{-s}$, absolutely convergent in suitable right half-planes, and satisfying a Hecke-type analytic functional equation

$$Q^s F(s) = \omega Q^{-s} \overline{G(-\overline{s})},$$

where $$F(s)=\phi(s) \prod_{i=1}^r \Gamma(\alpha_i s+\beta_i)$$, $L$0, and $L$1, $L$2. The system allows for multiple gamma factors in the archimedean parameter, generalizing previous work which required fewer constraints or only singly-parametrized gamma factors.

Main Theorems

1. Equivalence of Functional Equation and Modular Summation
   • The functional equation implies a modular summation identity:

   $L$3

   where $L$4 is an explicit integral kernel built from the gamma factors. - Conversely, the existence of this modular relation for arbitrary $L$5 guarantees the functional equation.

2. Riesz-Type Summation Identity and Equivalence

   • The functional equation implies a Riesz-type Voronoï summation formula for the partial sums of $L$6:

   $L$7

   where $L$8 and $L$9 are defined via Riesz means and contour integrals, and the right side involves explicit integral kernels. - This formula is also shown to be equivalent to the functional equation—a key new result of the paper.

3. Auxiliary Modular Relation as an Intermediate

   • The proof leverages a second, previously unconsidered, "auxiliary modular relation" built from Mellin transforms involving derivative gamma factors, demonstrating that this relation is equivalent to the functional equation and to the Voronoï/Riesz sum formula.
4. Existence of Multiple Characterizations
   • The main corollary: for Dirichlet series in this class,
     • The Hecke-type functional equation,
     • The modular-type summation relation,
     • The Riesz/Voronoï summation formula, and
     • The auxiliary modular relation are all logically equivalent.

Methodological Advances

The key theoretical innovation is the construction of an auxiliary modular relation, yielding a new route to deduce a functional equation from an analytic identity for the coefficients. This approach is enabled by precise analysis of Mellin transforms of products of gamma factors, asymptotics, and contour manipulation. Estimates for the associated kernel functions are given, and the essential necessity of the relaxation of growth conditions (compared to the classical Chandrasekharan–Narasimhan framework) is highlighted.

The argument controls uniform convergence and eliminates dependence on previously assumed stronger growth hypotheses, extending the scope of the equivalence.

Connections to the Literature

Previous work (Bochner, Chandrasekharan–Narasimhan, Berndt et al., Miller–Schmid, Kanemitsu–Tsukada) established the functional equation $L$0 Voronoï/Modular relation for narrower classes of $L$1-functions (e.g., with fewer gamma factors, or under stronger growth limits). This article encompasses the most general setting to date for the mutual equivalence in the presence of multiple gamma factors, including $L$2-functions not necessarily within the Selberg class.

Notable Implications

• Structural Characterization: The Voronoï/Riesz summation phenomenon, long observed as consequence of analytic continuation and functional equations, turns out to be in fact a characterization. Any arithmetic sequence $L$3 with such a formula must arise as coefficients of a Dirichlet series satisfying the required functional equation (with the explicit analytic archimedean kernel).
• Methodological Generality: The result applies to general automorphic and non-automorphic Dirichlet series with gamma factors, and subsumes numerous classical and modern summation identities (e.g., divisor functions, automorphic $L$4-functions for $L$5, $L$6, etc.), including those explored by Miller–Schmid and Goldfeld–Li.
• Foundation for Further Analysis: This equivalence forms a technical foundation for further study of subconvexity, oscillation, mean value theorems, and explicit moment computations for $L$7-functions. It supports the use of summation formulas as a proxy for functional equations within analytic and arithmetic applications.

Potential Future Directions

• Automorphic $L$8-Functions and Beyond: The formalism invites further work on modular-type relations for $L$9-functions of higher rank groups (e.g., $L$0 for $L$1) and potential applications to trace formulae.
• Characterization in Other Settings: The possibility exists to transfer similar equivalence arguments to $L$2-adic $L$3-functions, multiple Dirichlet series, and zeta functions outside the traditional Selberg class.
• Effective Computations: The explicit integral transforms and error analysis in the Riesz-type formulas provide avenues to refine numerical approximations and explicit formulas in arithmetic statistics.

Conclusion

This paper establishes a rigorous equivalence between the analytic functional equation (with multiple gamma factors), Voronoï-type summation identities, and modular relations for a large class of Dirichlet series. The work extends longstanding results in analytic number theory and offers a robust criterion for the arithmetic origin of sequences via summation formulas. Its methodological flexibility and generality position it as a substantial reference for further developments in the analysis of $L$4-functions and their applications in number theory and related fields (2604.02803).