1. Introduction

In number theory and many other branches of mathematics, distributions of finite sums of arithmetic functions constitute an active area of interest. An important tool for calculating the finite sum of a given arithmetic function is a summation formula of the form

(1.1)\begin{align} \sideset{}{'}\sum_{a\leq n\leq b}f(n)=\int_{a}^{b}f(x)\,dx+2\sum_{n=1}^{\infty}\int_{a}^bf(x)\cos(2\pi n x)\,dx. \end{align}

Here the function f can be taken as a function of bounded variation that is supported in a finite compact interval and the dash on the sum indicates that the last term of the sum is multiplied by $\frac{1}{2}$ if n = b. The Gauss circle problem and the Dirichlet divisor problem are two well-known problems in this direction. Dirichlet showed that if d(n) denotes the number of positive divisors of an integer n, then

\begin{align*} \Delta_d(x):=\sum_{n\leq x}d(n)-x\log x-(2\gamma -1)x=O(x^{1/2}). \end{align*}

On the other hand, if $r_2(n)$ is the number of ways to represent a positive integer n as a sum of squares of two integers then Gauss showed that

\begin{align*} \Delta_r(x)=\bigg\lvert\sum_{n\leq x}r_2(n)-\pi x\bigg\rvert=O(x^{1/2}). \end{align*}

Much later, in 1904, Voronoï [Reference Voronoï40, Reference Voronoï41] generalized the summation formula (1.1) to improve the bounds for $\Delta_d(x)$ and $\Delta_r(x)$ to $O(x^{1/3})$. In particular, he obtained in [Reference Voronoï40]

(1.2)\begin{align} \sideset{}{'}\sum_{a\leq n \leq b}d(n)f(n)=\int_a^bf(x)(\log x+2\gamma)\,dx+\sum_{n=1}^{\infty}d(n)\int_a^bf(x)L_0(4\pi\sqrt{nx})\,dx \end{align}

and

(1.3)\begin{align} \sideset{}{'}\sum_{a\leq n \leq b}r_2(n)f(n)=\pi\sum_{n=1}^{\infty}r_2(n)\int_a^bf(x)J_0(2\pi\sqrt{nx})\,dx, \end{align}

where f is a piecewise monotonic and continuous function, $L_0(x):=4K_0(x)-2\pi Y_0(x)$, K ₀ is the modified Bessel function of the second kind, and $J_0, Y_0$ are Bessel functions of the first and second kind respectively.

Voronoï summation is the best known method for improving the bounds for $\Delta_d(x)$ and $\Delta_r(x$) and was crucial in understanding the lower bounds of these two error terms. By using (1.2) and (1.3), Hardy and Landau observed in [Reference Hardy and Landau28] that $\Delta_d(x)$ and $\Delta_r(x)$ cannot be $O(x^{1/4})$. Consequently, they conjectured that both $\Delta_d(x)$ and $\Delta_r(x)$ are $O_\epsilon(x^{1/4+\epsilon})$ for any ϵ > 0.

Returning to the work in [Reference Voronoï40, Reference Voronoï41], Voronoï found a common pattern in the proofs of (1.2) and (1.3). This led him to propose analogous summation formulas of the type (1.2) and (1.3) for $\sum_{a\leq n\leq b}a(n)f(n)$, where a(n) is any ‘arithmetic function’ and f(n) is a ‘nice’ test function. Without evidence, it is generally accepted that a(n) is chosen to be the coefficient of some known L-function in order to produce analogues of the Voronoï summation formula. The choices for a(n) now also include automorphic forms as seen in [Reference Goldfeld and Li22], [Reference Miller and Schmid34], etc. The selection of the test function f(n) is an important component of the Voronoï summation formula. Ideally, one would like to choose discontinuous characteristic functions as candidates for f(n), since one is primarily interested in the partial sum $\sum_{a\leq n\leq b}a(n)$. However, one notes that smooth functions serve as good approximations to these test functions and due to the rapid decay of smooth functions, the integral transforms on the right-hand side of the summation formula (analogous to those on the right-hand side of (1.2) or (1.3)) can be analysed well. On the other hand, due to the approximation error, a small amount of information is lost when a characteristic function is replaced by a smooth function.

The main objective of this article is to revisit the coefficients a(n) in the Voronoï summation formula. We consider a broad class $\mathcal{C}$ of Dirichlet series satisfying functional equations of Hecke-type and derive analogues of the Voronoï summation formula for coefficients of these series. While results in this direction have been obtained in a lot of generality, the key aspect of this article is that we demonstrate the converse. More precisely, we show that if an ‘arithmetic function’ a(n) satisfies such a Voronoï-type summation formula, then it must appear as a coefficient of some Dirichlet series within this general class. In particular, we prove an equivalence between the functional equation of Hecke-type for the Dirichlet series corresponding to a(n) and the analogue of the Voronoï summation formula for a(n). While there are multiple results in various settings for the forward implication (that is, showing that the functional equation implies a summation formula), the converse is not known in much generality. We elaborate more upon this in §1. The converse is usually proved via an identity known as a modular relation (see for instance (1.20)). A unique feature of our result is that in order to prove this converse for our general L-function, we give an auxiliary modular relation which is different from the one occurring naturally for our Dirichlet series and has not appeared in the literature previously. Moreover, we show that this new modular relation is equivalent to the functional equation.

Modular identities have a rich history, going back to the ubiquitous Poisson summation formula, which serves as the basis for them. The Poisson summation formula is expressed as

\begin{align*} \sum_{n\in\mathbb{Z}}f(n)=\sum_{n\in\mathbb{Z}}\hat{f}(n) \end{align*}

for all Schwartz functions f. Here $\hat{f}(t):=\int_{\mathbb{R}}f(x)e^{-2\pi i t x}\,dx$ is the Fourier transform of f. If one considers the function $f(x)=e^{-\pi x^2 t}$, t > 0, then we have the well-known modular identity, also called the θ-identity:

\begin{align*} \theta(t):=\sum_{n\in\mathbb{Z}}e^{-\pi n^2 t}=\frac{1}{\sqrt{t}}\sum_{n\in\mathbb{Z}}e^{-\frac{\pi n^2}{t}}, \quad (t \gt 0). \end{align*}

It was based on this identity that Riemann [Reference Riemann36] in his landmark memoir built the theory of the zeta function and derived the functional equation of $\zeta(s)$. Results obtaining the θ-identity from the functional equation were given by Cahen [Reference Cahen14], Hamburger [Reference Hamburger23–Reference Hamburger26], and Hurwitz (unpublished note, cf. [Reference Oswald35]).

In order to state our results, we recall the Riesz-mean sum: $\sideset{}{'}\sum_{n\leq x}a(n)(x-n)^{\rho}$, with $\rho\ge 0$, where the dash on the sum indicates that the last term of the sum has to be multiplied by $\frac{1}{2}$ if ρ = 0 and x = n. After the work of Ramanujan on $\tau(n)$ (defined by $\sum_{n=1}^{\infty}\tau(n)z^n=z\prod_{n=1}^{\infty}(1-z^n)^{24}$ for $|z| \lt 1$), Wilton [Reference Wilton42] considered the Riesz sum $\sideset{}{'}\sum_{n\leq x}\tau(n)(x-n)^{\rho}$ and obtained the summation formula

(1.4)\begin{align} A_{\rho}(x):=\frac{1}{\Gamma(\rho +1)}\sideset{}{'}\sum_{n\leq x}\tau(n)(x-n)^{\rho}=\left(\frac{1}{2\pi}\right)^{\rho}\sum_{n=1}^{\infty}\left(\frac{x}{n}\right)^{6+\rho/2}\tau(n)J_{12+\rho}(4\pi\sqrt{nx}) \end{align}

for ρ > 0. Hardy [Reference Hardy27] improved this result and proved that ρ can be taken to be 0. This is the most desirable case. Recall that for any suitable smooth test function f, partial summation gives

(1.5)\begin{align} \sideset{}{'}\sum_{n\leq x}\tau(n)f(n)=A_0(x)f(x)-\int_1^xA_0(t)f'(t)\,dt. \end{align}

Substituting the infinite series representation of $A_0(t)$ from (1.4) into this and performing some elementary computations yields analogues of the Voronoï summation formula. Although the right-hand side of (1.4) converges absolutely and uniformly in any compact interval for large values of ρ, it can be seen to be convergent by some other means for small values of ρ. If we denote $S(x)=\sideset{}{'}\sum_{n\leq x} a(n)$ and $S_1(x)=\sideset{}{'}\sum_{n\leq x} a(n)(x-n)$ for a general arithmetic function a(n), then by partial summation, one has $S_1(x)=\int_1^x S(t)\, dt$. Since the infinite series representation of $S_1(x)$ (analogous to the right-hand side of (1.4)) converges rapidly and analysing the series is relatively simple, bounding S(x) in terms of $S_1(x)$ is fruitful. This is a key reason for studying Reisz sums. Moreover, if S(x) is monotonic then one can bound S(x) by the difference $S_1(x)-S_1(x(1\pm \delta))$ for some suitable choice of δ.

We now state our main results below. We will consider a class $\mathcal{C}$ of Dirichlet series with a more general functional equation with multiple Gamma factors. This class is defined as follows.

Given the functional equation (1.6), we define the vectors $\boldsymbol \alpha,\, \boldsymbol \beta$ and the quantity d_F by $\boldsymbol \alpha =(\alpha_1, \ldots, \alpha_r),\, \boldsymbol \beta=(\beta_1, \ldots, \beta_r)$, and $d_F=2\sum_{i=1}^{r}\alpha_i$. We will also denote the vector $(\bar{\beta}_1, \ldots, \bar{\beta}_r)$ by $\boldsymbol{\bar{\beta}}$. In the case of functions in the Selberg class, d_F is said to be the degree of the function, which acts as an invariant. We will denote $\frac{d_F}{2}$ by $d_F^{'}$ throughout this article. In [Reference Chandrasekharan and Narasimhan19], Chandrasekharan and Narasimhan considered the same class of Dirichlet series with the stronger restriction:

(1.8)\begin{align} \lim_{|t|\to\infty}\chi(\sigma+it)=0, \end{align}

uniformly in every bounded strip $-\infty \lt \sigma_1\leq\sigma\leq\sigma_2 \lt \infty$. In [Reference Chandrasekharan and Narasimhan19], they derived an identity of the type (1.13) and showed that if $\operatorname{Re} b(n)\neq 0$ for at least one value of n, then $ \operatorname{Re}[S_{\rho}(x)-Q_{\rho}(x)]= \Omega_{\pm}\left(x^{\theta}\right),$ where S_ρ and Q_ρ are as in (1.21) and (1.22), respectively, and $\theta=\frac{1}{d_F}\left(d_F^{'}\delta+\rho(d_F-1)-1/2\right)$.

The following two theorems are concerned with the equivalence of the functional equation and the modular relation for our class of Dirichlet series $\mathcal C$. Such equivalences have been proved by Bochner [Reference Bochner13] and Kanemitsu, Tanigawa, Tsukada [Reference Kanemitsu, Tanigawa and Tsukada30]. Results proving the modular relation from the functional equation have also been given by Tsukada [Reference Tsukada39] and Kanemitsu, Tanigawa, Tsukada [Reference Kanemitsu, Tanigawa and Tsukada29]. From each of these works, one can derive the corresponding result for a subset of functions in the class $\mathcal{C}$, namely those for which χ from definition 1.1 satisfies the stricter restriction (1.8). In theorems 1.2 and 1.3, we prove the equivalence of the functional equation and the modular relation for the entire class $\mathcal C$. In particular, we consider the more relaxed growth condition (1.7) on χ.

Theorem 1.2 Let $\phi(s)$ and $\psi(s)$ be two Dirichlet series in the class $\mathcal{C}$. Then the functional equation

(1.9)\begin{equation} Q^sF(s)=\omega Q^{{\delta}-s} \overline{G({\delta}-\bar{s})}, \end{equation}

where $F(s)= \phi(s)\prod_{i=1}^r \Gamma\left(\alpha_i s+\beta_i \right)$, $G(s) = \psi(s)\prod_{i=1}^r \Gamma\left(\alpha_i s+\beta_i \right)$, implies the modular relation

(1.10)\begin{equation} \sum_{n=1}^{\infty}a_nZ_{\boldsymbol\alpha, \boldsymbol \beta}(\lambda_nx)=P(x)+\frac{\omega}{(xQ)^\delta}\sum_{n=1}^{\infty}\bar{b}_n Z_{\boldsymbol \alpha,\boldsymbol{\bar{\beta}}}\bigg(\frac{\mu_n}{Q^2x}\bigg), \ \ (x \gt 0), \end{equation}

where for $x,a \gt 0$,

(1.11)\begin{equation} Z_{\boldsymbol\alpha,\boldsymbol\beta}(x):=\frac{1}{2 \pi i}\int_{a-i\infty}^{a+i\infty} \prod_{i=1}^{r} \Gamma(\alpha_i s+ \beta_i) x^{-s}ds. \end{equation}

P(x) is a residual function given by

(1.12)\begin{align} P(x)= \frac{1}{2 \pi i}\int_{C} F(s) x^{-s}ds, \end{align}

and C denotes a circle of finite radius, lying inside the domain D and containing all the singularities of F(s).

In the next result, we show that the functional equation implies a Riesz-sum identity for series in our class $\mathcal{C}$, with the smallest possible value of ρ, analogous to the ρ = 0 case in (1.4). Such results were proved by Chandrasekharan and Narasimhan [Reference Chandrasekharan and Narasimhan19, Eq. (1.3)] and Hafner [Reference Hafner43] for Dirichlet series satisfying functional equations with multiple Gamma factors, with the growth condition (1.8) on $\chi(s)$. Our result below holds for a larger class of functions, since χ fulfils the more relaxed growth condition (1.7).

Theorem 1.4 Let $\phi(s)$ and $\psi(s)$ be two Dirichlet series satisfying definition 1.1. Let $a \gt \max\{0,{\sigma}_a,{\sigma}_b\}$ be sufficiently large so that all the singularities of $F(s)= \phi(s)\prod_{i=1}^r \Gamma\left(\alpha_i s+\beta_i \right)$ are contained in the strip ${\delta}-a \lt \operatorname{Re}(s) \lt a$. Then the functional equation

\begin{equation*}Q^sF (s)=\omega Q^{\delta-s}\overline{G(\delta-\bar{s})},\end{equation*}

where $G(s) = \psi(s)\prod_{i=1}^r \Gamma\left(\alpha_i s+\beta_i \right)$, implies the identity

(1.13)\begin{align} & \frac{1}{\Gamma(\rho+1)}\sideset{}{'}\sum_{\lambda_n \leq x} a_{n}(x-\lambda_n)^\rho \notag\\ & \quad = Q_\rho(x)+\bigg(\frac{\omega x^{\delta+\rho}}{Q^\delta}\bigg) \sum_{n=1}^{\infty}\bar{b}_n\frac{1}{2 \pi i}\int_{a-i\infty}^{a+i\infty} \notag\\ & \qquad \times \frac{\Gamma(\delta-s) \prod_{i=1}^{r}\Gamma(\alpha_i s+\bar{\beta}_i)}{\Gamma(1+\delta-s+\rho)\prod_{i=1}^{r}\Gamma(\alpha_i(\delta-s)+\beta_i)}\bigg(\frac{\mu_nx}{Q^2}\bigg)^{-s}ds , \end{align}

for x > 0, and ${\rho} \gt (2{\sigma}_b-{\delta})d_F^{'}-\frac{1}{2}$. Here, $d_F^{'}=\sum_{i=1}^{r}\alpha_i$ and $Q_\rho(x)$ is a residual function given by

(1.14)\begin{equation} Q_\rho(x)=\frac{1}{2 \pi i}\int_{C}\frac{\phi(s)\Gamma(s)x^{s+\rho}}{\Gamma(s+\rho+1)} ds, \end{equation}

where $C=C_a$ denotes a circle of finite radius, lying inside the strip ${\delta}-a \lt \operatorname{Re}(s) \lt a$, containing all the singularities of F(s), such that all the singularities of $\phi(s)\Gamma(s)x^{s+\rho}$ which lie in this strip are contained inside C.

The main contribution of this article is to show that the Riesz sum identity (1.13) implies the functional equation. Our key idea is to prove this by introducing an auxiliary modular-type relation, different from the known modular relation (1.10). This new modular-type relation is given by

\begin{equation*} \sum_{n=1}^{\infty}a_nY_{\boldsymbol \alpha,\boldsymbol\beta}(\lambda_nx)=P_1(x)+\frac{\omega}{(xQ)^{\delta}}\sum_{n=1}^{\infty}\bar{b}_n X_{\boldsymbol\alpha,\boldsymbol{\bar{\beta}}}\bigg(\frac{\mu_n}{Q^2x}\bigg), \ \ x \gt 0, \end{equation*}

where $P_1(x)$ is a residual function which will be defined precisely later,

(1.15)\begin{equation} Y_{\boldsymbol \alpha,\boldsymbol\beta}(x):=\frac{1}{2 \pi i}\int_{a-i\infty}^{a+i\infty} \Gamma(s)\prod_{i=1}^{r} \Gamma(\alpha_i s+ \beta_i) x^{-s}ds, \qquad (x,a \gt 0) \end{equation}

and

(1.16)\begin{equation} X_{\boldsymbol\alpha,\boldsymbol{\bar{\beta}}}(x):=\frac{1}{2 \pi i}\int_{a-i\infty}^{a+i\infty} \Gamma(\delta-s)\prod_{i=1}^{r} \Gamma(\alpha_i s+ \bar{\beta_i}) x^{-s}ds, \qquad (x,a \gt 0). \end{equation}

It is this auxiliary modular-type relation which acts as an intermediate, allowing us to move from the Riesz sum identity (1.13) to the functional equation (1.9). In the following theorem, we first derive this new modular-type relation from the Riesz sum identity. Subsequently, we will deduce the functional equation (1.9) from this using similar techniques as in the proof of theorem 1.3.

Theorem 1.7 Let $a \gt \max\{0,{\sigma}_a,{\sigma}_b\}$ be sufficiently large so that all the singularities of $F(s)= \phi(s)\prod_{i=1}^r \Gamma\left(\alpha_i s+\beta_i \right)$ are contained in the strip ${\delta}-a \lt \operatorname{Re}(s) \lt a$. Let $C=C_a$ be a curve as in the statement of theorem 1.4. Then, the Riesz sum identity (1.13) implies the modular relation

(1.17)\begin{equation} \sum_{n=1}^{\infty}a_nY_{\boldsymbol \alpha,\boldsymbol\beta}(\lambda_nx)=P_1(x)+\frac{\omega}{(xQ)^{\delta}}\sum_{n=1}^{\infty}\bar{b}_n X_{\boldsymbol\alpha,\boldsymbol{\bar{\beta}}}\bigg(\frac{\mu_n}{Q^2x}\bigg), \ \ x \gt 0, \end{equation}

where $Y_{\boldsymbol \alpha,\boldsymbol\beta}(x)$ and $X_{\boldsymbol\alpha,\boldsymbol{\bar{\beta}}}(x)$ are as defined as in (1.15) and (1.16) respectively and $P_1(x)$ is a residual function given by

\begin{equation*} P_1(x)= \frac{1}{2 \pi i}\int_{C} \Gamma(s) F(s) x^{-s}ds. \end{equation*}

In the following theorem, we show that the functional equation (1.9) can be obtained from this auxiliary modular relation (1.17).

The following corollary follows from theorems 1.2, 1.3, 1.4, 1.7, and 1.8.

We now discuss in more detail recent literature related to these results. In [Reference Bochner13], Bochner defined Dirichlet series of Hecke-type as follows. Let λ_n and µ_n be two strictly increasing sequences of real numbers tending to $\infty$ and let ${a(n)}, {b(n)}$ be two sequences of complex numbers not identically zero. Consider the functions $\phi(s)$ and $\psi(s)$ representable as Dirichlet series

(1.18)\begin{equation} \phi(s)=\sum_{n=1}^{\infty} \frac{a(n)}{\lambda_n^{s}} \ \ \text{and}\,\,\, \psi(s)=\sum_{n=1}^{\infty} \frac{b(n)}{\mu_n^{s}}, \end{equation}

with finite abscissa of absolute convergence σ_a and σ_b respectively.

These Dirichlet series satisfy the functional equation

(1.19)\begin{align} (2\pi)^{-s} \Gamma(s)\phi(s)=(2\pi)^{s-\delta}\Gamma(\delta-s)\psi(\delta-s), \end{align}

for some δ > 0. That is, there exists a domain D which is the exterior of a bounded closed set S, such that in D there exists a holomorphic function $\chi(s)$ satisfying

\begin{align*} \lim_{|t|\to\infty}\chi(\sigma+it)=0 \end{align*}

uniformly in every bounded strip $-\infty \lt \sigma_1\leq\sigma\leq\sigma_2 \lt \infty$, with $\chi(s)=(2\pi)^{-s} \Gamma(s)\phi(s)$ for all $\sigma \gt \sigma_a$, and $\chi(s)=(2\pi)^{s-\delta}\Gamma(\delta-s)\psi(\delta-s)$ for all $\sigma \lt {\delta- \sigma_b}$. Bochner proved that the functional equation (1.19) is equivalent to the modular relation

(1.20)\begin{align} \sum_{n=1}^{\infty}a(n)e^{-\lambda_nx}=P(x)+\left(\frac{2\pi}{x}\right)^{\delta} \sum_{n=1}^{\infty}b(n)e^{-4\pi^2\mu_n/x}, \end{align}

where

\begin{align*} P(x)=\frac{1}{2\pi i}\int_{C}\chi(s)\left(\frac{2\pi}{x}\right)^s\,ds, \end{align*}

and C denotes a curve (or curves) in D containing the set S. Chandrasekharan and Narasimhan [Reference Chandrasekharan and Narasimhan17] considered the same functional equation with arbitrary real δ and a more relaxed growth condition on $\chi(s)$. In particular, they considered

\begin{align*} e^{-\epsilon|t|}\chi(\sigma+it)=O(1) \end{align*}

with $0 \lt \epsilon \lt \pi/2$ as $|t|\to\infty$, uniformly in every bounded strip $-\infty \lt \sigma_1\leq\sigma\leq\sigma_2 \lt \infty$. They proved the equivalence of the functional equation (1.19) and the modular relation (1.20). In the same article, they also considered the identity

(1.21)\begin{align} S_{\rho}(x) & := \frac{1}{\Gamma(\rho +1)}\sideset{}{'}\sum_{\lambda_n\leq x}a(n)(x-\lambda_n)^{\rho} \notag\\ & =\left(\frac{1}{2\pi}\right)^{\rho}\sum_{n=1}^{\infty}\left(\frac{x}{\mu_n}\right)^{\frac{1}{2}(\delta+\rho)}b(n)J_{\delta+\rho}(4\pi\sqrt{\mu_nx})+Q_{\rho}(x), \end{align}

where $J_{{\delta}+{\rho}}$ denotes the Bessel function of the first kind of order ${\delta}+{\rho}$, Q_ρ is a residual function defined by

(1.22)\begin{align} Q_{\rho}(x)=\frac{1}{2\pi i}\int_{C}\frac{\chi(s)(2\pi)^sx^{s+\rho}}{\Gamma(\rho+1+s)}\,ds, \end{align}

and C denotes a curve (or curves) in D containing the set S. They proved that the identity (1.21) for x > 0 and $\rho\geq 2\sigma_b-\delta-\frac{1}{2}$ is equivalent to the functional equation (1.19). In [Reference Chandrasekharan and Narasimhan18], the arithmetic identity (1.21) served as the main tool for Chandrasekharan and Narasimhan to obtain the omega-bound

\begin{align*} \operatorname{Re}[S_{\rho}(x)-Q_{\rho}(x)]=\Omega_{\pm}\left(x^{\frac{\delta}{2}+\frac{\rho}{2}-\frac{1}{4}}\right), \end{align*}

if $\operatorname{Re} b(n)\neq 0$ for at least one value of n and

\begin{align*} \operatorname{Im}[S_{\rho}(x)-Q_{\rho}(x)]=\Omega_{\pm}\left(x^{\frac{\delta}{2}+\frac{\rho}{2}-\frac{1}{4}}\right), \end{align*}

if $\operatorname{Im} b(n) \neq 0$ for at least one value of n.

Bochner (p. 341, [Reference Bochner13]) made the following observation about the modular relation (1.20). If the function $\chi(s)$ has simple poles at s = 0 and $s=\delta$ then the residual function P(x) can be written in the form $-a(0)+b(0)x^{-\delta}$ for some $a(0), b(0) \in \mathbb{C}$. In this case, the modular relation (1.20) can be re-written as

\begin{align*} \sum_{n=0}^{\infty}a(n)e^{-\lambda'_nx}={x}^{-\delta} \left( \sum_{n=0}^{\infty}b(n)e^{-\mu'_n/x} \right), \end{align*}

where $\lambda'_n={2\pi} {\lambda_n}$ and $\mu'_n={2\pi}{\mu_n}$ and $\lambda'_0 =0 =\mu_0'$. This modular relation implies the summation formula

(1.23)\begin{align} \sum_{n=0}^{\infty}a(n)f(\lambda'_nx)={x}^{-\delta} \sum_{n=0}^{\infty}b(n)g(\mu'_n/x), \end{align}

for some pair of functions $f,g$ which are connected by the Hankel transformations

\begin{align*} g(t)=t^{-\frac{1}{2}(\delta-1)}\int_0^{\infty}J_{\delta-1}(2\sqrt{tx})x^{\frac{1}{2}(\delta-1)}f(x)\,dx, \\ f(x)=x^{-\frac{1}{2}(\delta-1)}\int_0^{\infty}J_{\delta-1}(2\sqrt{tx})t^{\frac{1}{2}(\delta-1)}f(t)\,dt. \end{align*}

If one chooses x = 1 and the support of f is compact, then (1.23) yields analogues of the Voronoï summation formula (1.3). Moreover, if $\rho\geq 0$ (for instance if $2\sigma_b\geq \delta+ \frac{1}{2}$) in the identity (1.21), then analogues of the Voronoï summation formula can be obtained for $\sum a(n)f(n)$ by an argument similar to that discussed for (1.5).

In [Reference Berndt4], Berndt considered the Dirichlet series (1.18) and allowed for higher powers of the Gamma function in the functional equation. More precisely, he defined the holomorphic function $\chi(s)$ by $\chi(s)= \Gamma^m(s)\phi(s)$ for all $\sigma \gt \sigma_a$ and $\chi(s)=\Gamma^m(\delta-s)\psi(\delta-s)$ for all $\sigma \lt \delta-\sigma_b$. Let p be the smallest positive integer so that $\gamma= \sigma_b+p-\frac{1}{4m} \gt \max(0,\sigma_a,\sigma_b)$. Berndt showed that if

\begin{align*} \chi(s)=O\left(e^{\exp\left(\frac{\theta \pi |s|}{2\gamma-{\delta}}\right)}\right), \end{align*}

for some θ < 1, uniformly in the half-strip $\delta-\gamma \lt \sigma \lt \gamma$, $|t|\geq \eta$ contained in D, then we have an equivalence between the functional equation $\chi(s)=\chi({\delta}-s)$ and the identities

(1.24)\begin{align} \sum_{n=1}^{\infty}a(n)E_m(\lambda_nx)=P(x)+x^{-\delta} \sum_{n=1}^{\infty}b(n)E_m(\mu_n/x), \quad (x \gt 0) \end{align}

and

\begin{align*} & \frac{1}{\Gamma(\rho +1)}\sideset{}{'}\sum_{\lambda_n\leq x}a(n)(x-\lambda_n)^{\rho}\notag\\ & =2^{\rho(1-m)}\sum_{n=1}^{\infty}\left(\frac{x}{\mu_n}\right)^{\frac{1}{2}(\delta+\rho)}b(n)K_{\delta+\rho}(2^m\sqrt{\mu_nx}; {\delta}-1; m)+Q_{\rho}(x), \end{align*}

for $\delta \gt -1/2$ and $\rho \gt 2m\sigma_b-m\delta-1/2$. Here

\begin{align*} E_m(x)=\frac{1}{2\pi i}\int_{(c)}\Gamma^m(s)x^{-s}\,ds, \quad c \gt 0, x \gt 0, \end{align*}

\begin{align*} K_{\nu}(x;\mu;m)=\frac{1}{2\pi i } \int_{(c)}\frac{2^{2ms-(m-1)\mu-\nu-m+1}\Gamma(s)x^{\nu-2s}}{\Gamma^{m-1}(\mu+1-s)\Gamma(\nu+1-s)}\,ds,\quad 0 \lt c \lt \frac{1}{2}\min (\mu,\nu)+\frac{3}{4}, \end{align*}

\begin{align*} P(x)=\frac{1}{2\pi i}\int_{C}\chi(s)x^{-s}\,ds \ \text{and } Q_{\rho}(x)=\frac{1}{2\pi i}\int_{C}\frac{\Gamma(s)\phi(s)x^{s+\rho}}{\Gamma(\rho+1+s)}\,ds, \end{align*}

where C denotes a curve (or curves) in D containing the set S.

In a series of articles, Berndt [Reference Berndt3, Reference Berndt5–Reference Berndt8] presented a detailed study of Riesz-type summation formulae for various classes of Dirichlet series. In [Reference Berndt, Kim and Zaharescu12], Berndt, Kim, and Zaharescu studied various Riesz-type sums and then in [Reference Kim32], Kim gave Riesz-type identities for weighted sums of divisor functions. In recent work by Berndt, Dixit, Gupta, and Zaharescu [Reference Berndt, Dixit, Gupta and Zaharescu9, Reference Berndt, Dixit, Gupta and Zaharescu10], they obtained new identities of the type (1.21) involving the modified Bessel function $K_{\nu}(z)$ of order ν.

In [Reference Berndt, Dixit, Roy and Zaharescu11, theorem 6.1], the first author along with Berndt, Dixit, and Zaharescu proved a Voronoï summation formula analogous to (1.2) for the divisor function $\sigma_s(n)$. Subsequently, Dixit, Maji, and Vatwani [Reference Dixit, Maji and Vatwani21, theorem 2.2] obtained a version of the Voronoï summation formula for a more general divisor function $\sigma_s^{(k)}(n)$, which gives [Reference Berndt, Dixit, Roy and Zaharescu11, theorem 6.1] as a corollary but for a larger domain. Again in [Reference Banerjee and Maji2], Banerjee and Maji obtained a summation formula for $\sigma_s^{(k)}(n)$ involving the modified Bessel function of the second kind.

In the literature, there are many studies relevant to obtaining the modular-type relation (1.20) from the functional equation in various settings. For instance, Arai, Chakraborty, Kanemitsu [Reference Arai, Chakraborty and Kanemitsu1] and Chakraborty, Kanemitsu, Maji [Reference Chakraborty, Kanemitsu and Maji15] obtained modular-type relations associated to Dedekind zeta functions and the Rankin–Selberg L-function respectively. A breakthrough result of Tsukada [Reference Tsukada39] gives a more generalized modular-type identity for a class of Dirichlet series. This general result allows one to recover many existing modular-type identities as special cases, for instance (1.20) and (1.24). Further in [Reference Kanemitsu, Tanigawa and Tsukada29, Reference Kanemitsu, Tanigawa and Tsukada30], Kanemitsu, Tanigawa, and Tsukada studied these kinds of generalized modular relations along with many number theoretic applications. In [Reference Kanemitsu, Tanigawa and Tsukada30], they established the equivalence between the functional equation and modular relation in a very general setting. For a more detailed description of these generalized modular relations, we refer the reader to the book [Reference Kanemitsu and Tsukada31] by Kanemitsu and Tsukada.