Some New Applications of the Mellin Transform Involving the Lambert Transforms and Implications for the Riemann Hypothesis

Abstract

This work investigates the interplay between the Mellin transform and Lambert transforms to derive several novel results. In particular, we establish new inversion formulae for the Lambert transforms along with a Plancherel-type identity. Additionally, we explore the implications of these findings, highlighting their relevance to Salem’s equivalence and potential connections with the Riemann hypothesis.

1. Introduction and Preliminaries

The Mellin transform and its interplay with Lambert-type transforms have become central tools in analytic number theory and integral transform analysis. Recall that the Mellin transform of a function f is defined by

$$\left(\mathcal{M}\left[f\right]\right)\left(s\right)={\int }_0^{\infty }f\left(t\right)t^{s-1}dt,s\in \mathbb{C},$$

(1)

whenever the integral converges. Its inversion and Parseval-type properties form a foundational element in complex analysis and are closely tied to the distribution of zeros of the Riemann zeta function.

A classical Lambert series is given by

$$\sum _{k=1}^{\infty }{\alpha }_k{\displaystyle \frac{x^k}{1-x^k}},\left|x\right|<1,{\alpha }_k\in \mathbb{C}.$$

This series generalizes identities first studied by Lambert [1]. For example, the choices ${\alpha }_k=1$ and ${\alpha }_k=k$ yield

$$\sum _{k=1}^{\infty }{\displaystyle \frac{x^k}{1-x^k}}=\sum _{k=1}^{\infty }\tau \left(k\right)x^k,\sum _{k=1}^{\infty }k{\displaystyle \frac{x^k}{1-x^k}}=\sum _{k=1}^{\infty }\sigma \left(k\right)x^k,\left|x\right|<1,$$

where $\tau \left(k\right)$ and $\sigma \left(k\right)$ denote the divisor function and sum-of-divisors function, respectively. These identities illustrate how Lambert series encode multiplicative arithmetic information in a generating framework.

Motivated by the structure of these series, the Lambert transform was introduced as a continuous analogue [2]:

$$\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)={\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}-1}}dt,x>0,\delta \in \mathbb{R},$$

(2)

for functions f ensuring absolute convergence of the integral. Since its introduction, the Lambert transform has been extensively studied regarding convergence, asymptotic behaviour, inversion formulae, and structural generalizations. Miller [3,4] analyzed summability and convergence criteria essential for inversion theory, while Ferreira and López [5] studied boundary behaviour. Refinements of Widder’s classical inversion formula, often via Möbius-type limiting processes, have been obtained in [6,7,8]. Further developments connect Lambert transforms with the Hurwitz and generalized Riemann zeta functions [9,10] and formulations in terms of Stieltjes integral kernels [11], highlighting their analytic and arithmetic significance. The present paper systematically explores the connection between the Mellin and Lambert transforms. Section 1 introduces the relevant definitions and notation. Section 2 establishes a novel relation between these transforms, enabling the derivation of an inversion formula for the Lambert transform via the classical Titchmarsh inversion theorem for the Mellin transform. In Section 3, we prove an $L^p$$(1<p\le 2)$ inversion theorem, a Plancherel-type identity, and a corresponding uniqueness result. Section 4 extends these results to generalized Lambert transforms, refining earlier studies [11,12,13,14,15,16], while Section 5 concludes with observations and potential directions for future research.

The Riemann hypothesis, one of mathematics’ most profound open problems, admits several reformulations across analytic number theory, harmonic analysis, and operator theory. The Riemann hypothesis establishes a deep link between the analytic properties of the Riemann zeta function and the distribution of prime numbers. It has inspired transform-based formulations where the location of zeros corresponds to specific functional conditions. Such connections motivate the study of generalized transforms whose structural properties parallel those underlying the hypothesis. Notably, Salem [17] established a criterion linking the boundedness of certain integrals involving Fourier coefficients to the Riemann hypothesis. Recall that the Riemann zeta function is defined by

$$\zeta \left(s\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{1}{k^s}},\Re \left(s\right)>1,$$

and extends analytically via the Dirichlet eta function

$$\eta \left(s\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{{(-1)}^{k-1}}{k^s}},\Re \left(s\right)>0,$$

satisfying

$$\eta \left(s\right)=(1-2^{1-s})\zeta \left(s\right),\Re \left(s\right)>1.$$

This analytic continuation allows the study of $\zeta \left(s\right)$ in the critical strip $0<\Re \left(s\right)<1$. Salem’s equivalence [17,18] asserts that the Riemann hypothesis is true if and only if specific boundedness conditions involving integral transforms are satisfied within the critical strip, emphasizing the critical role of transform methods in analytic number theory.

Throughout this work, we employ weighted Lebesgue spaces $L_{a,p}\left({\mathbb{R}}_{+}\right)$, $a\in \mathbb{R}, 1\le p<\infty$, consisting of measurable complex-valued functions with the norm

$${\parallel f\parallel }_{a,p}={\left({\int }_0^{\infty }{\left|f\left(t\right)\right|}^p{t}^{ap-1}dt\right)}^{1/p}.$$

These spaces provide a natural functional setting to rigorously establish inversion and Plancherel-type results.

2. Mellin Versus Lambert Transforms

This section explores a significant relation between the Lambert transform (2) and the Mellin transform (1).

Consider $s\in \mathbb{C}$ such that $\Re s<\delta$, and let $t>0$. Then,

$$\begin{array}{c}{\int }_0^{\infty }{x}^{-s}{\displaystyle \frac{x^{\delta -1}}{e^{xt}-1}}dx={\int }_0^{\infty }{x}^{-s+\delta -1}\left({\displaystyle \sum _{k=1}^{\infty }}{e}^{-kxt}\right)dx\hfill \\ ={\int }_0^{\infty }{\displaystyle \underset{n\to \infty }{lim}}\left({\displaystyle \sum _{k=1}^n}{x}^{-s+\delta -1}{e}^{-kxt}\right)dx.\hfill \end{array}$$

(3)

For any $n\in \mathbb{N}$, we estimate the absolute value of the summation:

$$\begin{array}{ccc}& & \sum _{k=1}^n\left|x^{-s+\delta -1}{e}^{-kxt}\right|=\sum _{k=1}^n{x}^{-\Re s+\delta -1}{e}^{-kxt}\le {\displaystyle \frac{x^{-\Re s+\delta -1}}{e^{xt}-1}}.\hfill \end{array}$$

Since the inequality ${\displaystyle \frac{xt}{e^{xt}-1}}\le 1$ holds for all $x,t>0$, the function ${\displaystyle \frac{x^{-\Re s+\delta -1}}{e^{xt}-1}}$ is integrable over ${\mathbb{R}}_{+}$ when $\Re s<\delta -1$. Thus, by the dominated convergence theorem, we may interchange limit and integration to obtain that (3) is equal to

$$\sum _{k=1}^{\infty }{\int }_0^{\infty }{x}^{-s+\delta -1}{e}^{-kxt}dx,\mathrm{for}\Re s<\delta -1.$$

Using the substitution $u=kxt$, we compute:

$$\begin{array}{ccc}& & \sum _{k=1}^{\infty }{\int }_0^{\infty }{\displaystyle \frac{u^{-s+\delta -1}}{{\left(kt\right)}^{-s+\delta }}}{e}^{-u}du=\sum _{k=1}^{\infty }{\displaystyle \frac{t^{s-\delta }}{k^{s-\delta }}}\Gamma (\delta -s)\hfill \\ & & =t^{s-\delta }\Gamma (\delta -s)\sum _{k=1}^{\infty }{\displaystyle \frac{1}{k^{\delta -s}}}=t^{s-\delta }\Gamma (\delta -s)\zeta (\delta -s),\mathrm{for}\Re s<\delta -1.\hfill \end{array}$$

Now, suppose $f\in L_{\Re s,1}\left({\mathbb{R}}_{+}\right)$ and $\Re s<\delta -1$. Applying Fubini’s theorem, we obtain:

$$\begin{array}{ccc}& & {\int }_0^{\infty }{t}^{s-1}f\left(t\right)\Gamma (\delta -s)\zeta (\delta -s)dt={\int }_0^{\infty }{t}^{\delta -1}f\left(t\right)·t^{s-\delta }\Gamma (\delta -s)\zeta (\delta -s)dt\hfill \\ & & ={\int }_0^{\infty }{t}^{\delta -1}f\left(t\right){\int }_0^{\infty }{x}^{-s}{\displaystyle \frac{x^{\delta -1}}{e^{xt}-1}}dxdt={\int }_0^{\infty }{x}^{-s}{\int }_0^{\infty }{t}^{\delta -1}f\left(t\right){\displaystyle \frac{x^{\delta -1}}{e^{xt}-1}}dtdx\hfill \\ & & ={\int }_0^{\infty }{x}^{-s+\delta -1}{\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}-1}}dtdx=\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s),\hfill \end{array}$$

where $\mathcal{M}$ denotes the Mellin transform defined in (1).

Since $\Gamma (\delta -s)$ and $\zeta (\delta -s)$ are nonzero in the region $\Re s<\delta -1$, we conclude that:

$$\left(\mathcal{M}\left[f\right]\right)\left(s\right)={\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)\zeta (\delta -s)}},\mathrm{for}\Re s<\delta -1.$$

(4)

These findings are encapsulated in the following theorem.

Theorem 1.

Let $\delta \in \mathbb{R}$. Suppose f is a measurable function on ${\mathbb{R}}_{+}$ such that the integral defining the transform (2) exists, and assume $f\in L_{\Re s,1}\left({\mathbb{R}}_{+}\right)$. Then,

$$\left(\mathcal{M}\left[f\right]\right)\left(s\right)={\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)\zeta (\delta -s)}},\mathrm{for}\Re s<\delta -1.$$

Now, from Theorem 1 and using the Titchmarsh inversion formula for the Mellin transform (see [19], Theorem 28, p. 46), we derive the following result:

Theorem 2.

Let $\delta \in \mathbb{R}$. Assume that f is a measurable function on ${\mathbb{R}}_{+}$ for which the integral in (2) converges, and suppose that $t^{a-1}f\left(t\right)\in L^1\left({\mathbb{R}}_{+}\right)$ for some $a<\delta -1$. Furthermore, let f be of bounded variation in a neighborhood of each $t>0$. Then

$${\displaystyle \frac{1}{2}}\{f(t+0)+f(t-0)\}={\displaystyle \frac{1}{2\pi i}}\underset{T\to +\infty }{lim}{\int }_{a-iT}^{a+iT}{\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)\zeta (\delta -s)}}{t}^{-s}\mathrm{d}s.$$

Corollary 1.

Under the assumptions of Theorem 2, if f is also continuous at $t>0$, then

$$f\left(t\right)=\underset{T\to +\infty }{lim}{\int }_{a-iT}^{a+iT}{\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)\zeta (\delta -s)}}{t}^{-s}\mathrm{d}s.$$

Corollary 2.

Let $\delta \in \mathbb{R}$ and suppose f satisfies the hypotheses of Theorem 2, is continuous on ${\mathbb{R}}_{+}$, and is of bounded variation near every point of ${\mathbb{R}}_{+}$. Then

$$\mathrm{if}{\mathcal{L}}_{\delta }\left[f\right]=0\mathrm{almost}\mathrm{everywhere}\mathrm{on}{\mathbb{R}}_{+},\mathrm{then}f\equiv 0.$$

3. An ${\mathit{L}}^{\mathit{p}}$ Inversion Formula and a Plancherel-Type Theorem

By utilizing the $L^p$ inversion formula for the Mellin transform, valid for $1<p\le 2$ (see [19]), along with the relationship (4) between the Mellin transform and the integral transform defined in (2), we derive an $L^p$ inversion formula for the Lambert transform (2).

As stated in [20], p. 694, for any function $f\in L_{a,p}\left({\mathbb{R}}_{+}\right)$, with $1<p\le 2$, the Mellin transform is defined by

$$\left(\mathcal{M}\left[f\right]\right)\left(s\right)={\int }_0^{\infty }f\left(t\right)t^{s-1}\mathrm{d}t,(\Re s=a),$$

(5)

where the integral is convergent in the mean with respect to the norm on $L_q\left((a-i\infty , a+i\infty )\right)$, with $q=p/(p-1)$.

The inverse Mellin transform is given by

$$f\left(t\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{a-i\infty }^{a+i\infty }\left(\mathcal{M}\left[f\right]\right)\left(s\right)t^{-s}\mathrm{d}s,$$

and the integral converges in the mean with respect to the norm in $L_{a,p}\left({\mathbb{R}}_{+}\right)$.

If $f\in L_{a,1}\left({\mathbb{R}}_{+}\right)\cap L_{a,p}\left({\mathbb{R}}_{+}\right)$, then (5) agrees almost everywhere with the classical Mellin transform (1); see [20] for further details.

Based on the above and results from Titchmarsh [19] and using relation (4), we obtain the following:

Theorem 3.

Let $\delta \in \mathbb{R}$. Suppose that f is a measurable function on ${\mathbb{R}}_{+}$ such that the integral

$$\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)={\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}-1}}dt,x>0,$$

converges, and that $f\in L_{a,1}\left({\mathbb{R}}_{+}\right)\cap L_{a,p}\left({\mathbb{R}}_{+}\right)$ for some $1<p\le 2$ and $a<\delta -1$. Then,

$$\underset{T\to \infty }{lim}{\int }_0^{\infty }{|f\left(t\right)-f(t,T)|}^p{t}^{ap-1}\mathrm{d}t=0,$$

where

$$f(t,T)={\displaystyle \frac{1}{2\pi i}}{\int }_{a-iT}^{a+iT}{\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)\zeta (\delta -s)}}{t}^{-s}\mathrm{d}s.$$

Corollary 3.

Under the assumptions of Theorem 3, if ${\mathcal{L}}_{\delta }\left[f\right]=0$ almost everywhere on ${\mathbb{R}}_{+}$, then $f=0$ almost everywhere on ${\mathbb{R}}_{+}$.

Therefore, one has

Corollary 4.

Let f be a bounded measurable function such that $f\left(t\right)=O\left(t^{1/2}\right)$ as $t\to 0^{+}$. Let ${\displaystyle \frac{1}{2}}<\delta <1$, and assume that f satisfies the homogeneous integral equation

$${\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}-1}}dt=0,x>0,$$

then, $f=0$ almost everywhere on ${\mathbb{R}}_{+}$.

Proof. 

Since ${\displaystyle \frac{1}{2}}<\delta <1$, it follows that ${\displaystyle \frac{1}{2}}=1-{\displaystyle \frac{1}{2}}>1-\delta$. Choosing a such that ${\displaystyle \frac{1}{2}}>-a>1-\delta$, the given class of functions meets the requirements of Theorem 3, and the conclusion follows. □

Using the relationship (4) established between the Lambert transform and the Mellin transform, we can derive a Plancherel-type result for the Lambert transform (2).

By applying Titchmarsh’s Plancherel theorem for the Mellin transform ([19], Theorem 71, pp. 94–95) and using Theorem 1, we arrive at the following result:

Theorem 4 (Plancherel-type theorem).

Let $\delta \in \mathbb{R}$. Suppose that f is a measurable function on ${\mathbb{R}}_{+}$ such that the integral

$$\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)={\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}-1}}dt,x>0,$$

converges, and that $f\in L_{a,1}\left({\mathbb{R}}_{+}\right)\cap L_{a,2}\left({\mathbb{R}}_{+}\right)$ for some $a<\delta -1$. Then

$${\int }_0^{\infty }{\left|f\left(x\right)\right|}^2{x}^{2a-1}dx={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\left|{\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -a-it)}{\Gamma (\delta -a-it)\zeta (\delta -a-it)}}\right|}^{2}dt.$$

(6)

Proof. 

Since $f\in L_{a,2}\left({\mathbb{R}}_{+}\right)$, the Titchmarsh’s Plancherel theorem for the Mellin transform ([19], Theorem 71) yields to

$${\int }_0^{\infty }{\left|f\left(x\right)\right|}^2{x}^{2a-1}dx={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\left|\left(\mathcal{M}\left[f\right]\right)(a+it)\right|}^{2}dt.$$

Because $f\in L_{a,1}\left({\mathbb{R}}_{+}\right)$, the transform $\mathcal{M}$ matches the standard Mellin transform (1) almost everywhere on ${\mathbb{R}}_{+}$. Now, by applying Theorem 1, we obtain identity (6). □

Corollary 5.

Under the assumptions of Theorem 4, we have:

$$\mathrm{If}{\mathcal{L}}_{\delta }\left[f\right]=0\mathrm{almost}\mathrm{everywhere}\mathrm{on}{\mathbb{R}}_{+},\mathrm{then}f=0\mathrm{almost}\mathrm{everywhere}\mathrm{on}{\mathbb{R}}_{+}.$$

Proof. 

If ${\mathcal{L}}_{\delta }\left[f\right]=0$ almost everywhere on ${\mathbb{R}}_{+}$, then the right-hand side of (6) vanishes, implying:

$${\int }_0^{\infty }{\left|f\left(x\right)\right|}^2{x}^{2a-1}dx=0,$$

and thus $f=0$ almost everywhere on ${\mathbb{R}}_{+}$. □

We now again address the next result

Corollary 6.

Let f be a bounded measurable function satisfying $f\left(t\right)=O\left(t^{1/2}\right)$ as $t\to 0^{+}$. Suppose ${\displaystyle \frac{1}{2}}<\delta <1$ and that f solves the homogeneous integral equation

$$\begin{array}{c}\hfill {\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}-1}}dt=0,x>0,\end{array}$$

(7)

then $f=0$ almost everywhere on ${\mathbb{R}}_{+}$.

Proof. 

Since ${\displaystyle \frac{1}{2}}<\delta <1$, we have ${\displaystyle \frac{1}{2}}=1-{\displaystyle \frac{1}{2}}>1-\delta$. Choose a such that ${\displaystyle \frac{1}{2}}>-a>1-\delta$. Then f satisfies the hypotheses of Theorem 4, and the result follows. □

Remark 1.

Observe that the result obtained in Corollary 6 (which agrees with Corollary 4) is readily extended to the following:

Let f be a bounded measurable function on ${\mathbb{R}}_{+}$ such that $f\left(t\right)=O\left(t^{1-b}\right)$ as $t\to 0^{+}$ and set $b<\delta <1$. Assume that f is a solution of the homogeneous integral Equation (7), then $f=0$ almost everywhere on ${\mathbb{R}}_{+}$.

In order to prove this result, one takes a such that $1-b>-a>1-\delta$ and one translates the proof of Corollary 6 to this setting.

4. Some Considerations on Related Generalized Lambert Transforms

In this section we analyze two generalized Lambert transforms by means of the results obtained in the previous sections. One of them has a hyperbolic kernel and the other is the Widder–Lambert transform.

4.1. A Generalized Lambert Transform with Hyperbolic Kernel

We consider the integral transform with the hyperbolic kernel of a suitable complex-valued function f on ${\mathbb{R}}_{+}$ by means of

$$\begin{array}{c}\hfill \left({\tilde{\mathcal{L}}}_{\delta }\left[f\right]\right)\left(x\right)={\int }_0^{\infty }f\left(t\right)t^{\delta -1}{\displaystyle \frac{cosech\left(xt\right)}{2}}dt,x>0,\delta \in \mathbb{R}.\end{array}$$

(8)

When $\delta =1$, the kernel $cosech\left(xt\right)={\displaystyle \frac{1}{e^{xt}-e^{-xt}}}$ is written as

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{e^{xt}-e^{-xt}}}={\displaystyle \frac{e^{xt}}{e^{2xt}-1}}=\sum _{k=1}^{\infty }{a}_k{e}^{-kxt},\end{array}$$

where $a_{2k-1}=1$ and $a_{2k}=0$, $k=1,2,3,\dots$, whereof (8) is a generalized Lambert transform in the sense of [11]. This transform was considered in [14] in order to obtain an inversion formula for the Stieltjes transform. In [15] this transform was also considered in order to obtain an inversion formula for the Stieltjes–Poisson transform.

Now, observe that

$$\begin{array}{c}\hfill {\displaystyle \frac{e^x}{e^{2x}-1}}={\displaystyle \frac{1}{e^x-1}}-{\displaystyle \frac{1}{e^{2x}-1}},x>0.\end{array}$$

Thus, when the Lambert transform $\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)$ given by (2) converges for $x>0$ then one has

$$\begin{array}{c}\hfill \left({\tilde{\mathcal{L}}}_{\delta }\left[f\right]\right)\left(x\right)=\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)-\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(2x\right).\end{array}$$

(9)

On the other hand

$$\begin{array}{c}\hfill \left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\left(2x\right)\right]\right)\left(s\right)=2^{-s}\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\left(x\right)\right]\right)\left(s\right),s\in \mathbb{C},\end{array}$$

(10)

when the integrals exist and $\mathcal{M}$ being the Mellin transform given by (1).

Thus, from (9) and (10) one has

$$\begin{array}{c}\hfill \left(\mathcal{M}\left[{\tilde{\mathcal{L}}}_{\delta }\left[f\right]\right]\right)(\delta -s)=\left(1-2^{s-\delta }\right)\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\left(x\right)\right]\right)(\delta -s),s\in \mathbb{C},\delta \in \mathbb{R},\end{array}$$

(11)

when the integrals exist.

Now, since for $f\in L_{\Re s,1}\left({\mathbb{R}}_{+}\right)$, $\Re s<\delta -1,$$\delta \in \mathbb{R}$, and since $1-2^{s-\delta }\ne 0$ in $\Re s<\delta -1,$ then the expressions (4) and (11) yield

$$\begin{array}{c}\hfill \left(\mathcal{M}\left[f\right]\right)\left(s\right)={\displaystyle \frac{\left(\mathcal{M}\left[{\tilde{\mathcal{L}}}_{\delta }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)(1-2^{s-\delta })\zeta (\delta -s)}}.\end{array}$$

(12)

Now, using the relation (12), one obtains the corresponding results to those in Section 2 and Section 3 for the transform ${\tilde{\mathcal{L}}}_{\delta }$.

Specifically, Corollary 4 becomes

Corollary 7.

Let f be a bounded measurable function such that $f\left(t\right)=O\left(t^{1/2}\right)$ as $t\to 0^{+}$. Suppose ${\displaystyle \frac{1}{2}}<\delta <1$, and that f solves the homogeneous integral equation

$${\int }_0^{\infty }f\left(t\right)t^{\delta -1}cosech\left(xt\right)dt=0,x>0,$$

then $f=0$ almost everywhere on ${\mathbb{R}}_{+}$.

4.2. The Widder–Lambert Transform

Observe that when one considers the Widder–Lambert transform of a suitable complex-valued function f on ${\mathbb{R}}_{+}$ given by

$$\begin{array}{c}\hfill \left({\mathcal{W}}_{\delta }\left[f\right]\right)\left(x\right)={\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}+1}}dt,x>0,\delta \in \mathbb{R},\end{array}$$

(13)

(see [12,13,15]) and taking into account the fact that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{e^x+1}}={\displaystyle \frac{1}{e^x-1}}-{\displaystyle \frac{2}{e^{2x}-1}},x>0,\end{array}$$

one arrives at

$$\begin{array}{c}\hfill \left({\mathcal{W}}_{\delta }\left[f\right]\right)\left(x\right)=\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)-2\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(2x\right),x>0.\end{array}$$

Then one has

$$\begin{array}{c}\hfill \left(\mathcal{M}\left[{\mathcal{W}}_{\delta }\left[f\right]\right]\right)(\delta -s)=\left(1-2^{1+s-\delta }\right)\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s),s\in \mathbb{C},\delta \in \mathbb{R},\end{array}$$

when the integrals exist.

Now, since for $f\in L_{\Re s,1}\left({\mathbb{R}}_{+}\right)$, $\Re s<\delta -1,$$\delta \in \mathbb{R}$, and since $1-2^{1-(\delta -s)}\zeta (\delta -s)=\eta (\delta -s)\ne 0$ in $\Re s<\delta -1,$ then one arrives at

$$\begin{array}{c}\hfill \left(\mathcal{M}\left[f\right]\right)\left(s\right)={\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{W}}_{\delta }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)\eta (\delta -s)}}.\end{array}$$

(14)

Note that the relation (14) agrees with Equation (2.3) in [12].

Corollary 4 becomes:

Let f be a bounded measurable function such that $f\left(t\right)=O\left(t^{1/2}\right)$ as $t\to 0^{+}$. Suppose ${\displaystyle \frac{1}{2}}<\delta <1$, and that f solves the homogeneous integral equation

$${\int }_0^{\infty }f\left(t\right){\displaystyle \frac{t^{\delta -1}}{e^{xt}+1}}dt=0,x>0,$$

then $f=0$ almost everywhere on ${\mathbb{R}}_{+}$.

Observe that this result agrees with Corollary 3.3 in [12] which is an approach to the Salem equivalence to the Riemann hypothesis (also see [13,16]).

Remark 2 (A unified extension).

The results of the previous sections are readily extended to the integral transform

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{\delta ,\alpha }\left[f\right]\right)\left(x\right)={\int }_0^{\infty }f\left(t\right)t^{\delta -1}{\displaystyle \frac{e^{xt}+\alpha }{e^{2xt}-1}}dt,x>0,\delta \in \mathbb{R},-1\le \alpha \le 3.\end{array}$$

(15)

Observe that when $\delta =1$, the kernel of (15) becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{e^{xt}+\alpha }{e^{2xt}-1}}=\sum _{k=1}^{\infty }{a}_k{e}^{-kxt},\end{array}$$

where $a_{2k-1}=1$ and $a_{2k}=\alpha$, $k=1,2,3,\dots$, whereof (15) is a generalized Lambert transform in the sense of [11].

Note that when $\alpha =1$ the transform (15) becomes the Lambert transform ${\mathcal{L}}_{\delta }\left[f\right]$ given by (2). Also, when $\alpha =0$, the transform (15) becomes the transform ${\tilde{\mathcal{L}}}_{\delta }\left[f\right]$ given by (8), and when $\alpha =-1$ the transform (15) becomes the Widder–Lambert transform ${\mathcal{W}}_{\delta }\left[f\right]$ given by (13).

Now, observe that

$$\begin{array}{c}\hfill {\displaystyle \frac{e^x+\alpha }{e^{2x}-1}}={\displaystyle \frac{1}{e^x-1}}-{\displaystyle \frac{(1-\alpha )}{e^{2x}-1}},x>0,\alpha \in \mathbb{R}.\end{array}$$

Thus, when $\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)$ converges for $x>0$ then one has

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{\delta ,\alpha }\left[f\right]\right)\left(x\right)=\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(x\right)-(1-\alpha )\left({\mathcal{L}}_{\delta }\left[f\right]\right)\left(2x\right).\end{array}$$

(16)

Thus, from (16) and (10) one has

$$\begin{array}{c}\hfill \left(\mathcal{M}\left[{\mathcal{L}}_{\delta ,\alpha }\left[f\right]\right]\right)(\delta -s)=\left(1-(1-\alpha )2^{s-\delta }\right)\left(\mathcal{M}\left[{\mathcal{L}}_{\delta }\left[f\right]\right]\right)(\delta -s),s\in \mathbb{C},\delta \in \mathbb{R},\end{array}$$

(17)

when the integrals exist.

Note that for $-1\le \alpha \le 3,$ the expression $1-(1-\alpha )2^{s-\delta }$ is not zero in $\Re s<\delta -1,$$\delta \in \mathbb{R}$.

So, for $f\in L_{\Re s,1}\left({\mathbb{R}}_{+}\right)$ and $\Re s<\delta -1,$ Equations (4) and (17) yield

$$\begin{array}{c}\hfill \left(\mathcal{M}\left[f\right]\right)\left(s\right)={\displaystyle \frac{\left(\mathcal{M}\left[{\mathcal{L}}_{\delta ,\alpha }\left[f\right]\right]\right)(\delta -s)}{\Gamma (\delta -s)(1-(1-\alpha )2^{s-\delta })\zeta (\delta -s)}},\delta \in \mathbb{R},-1\le \alpha \le 3.\end{array}$$

(18)

Now, using the relation (18), one obtains the corresponding results to those in Section 2 and Section 3 for the extended transform ${\mathcal{L}}_{\delta ,\alpha },\delta \in \mathbb{R},-1\le \alpha \le 3$.

Specifically, Corollary 4 becomes

Set $-1\le \alpha \le 3.$ Let f be a bounded measurable function such that $f\left(t\right)=O\left(t^{1/2}\right)$ as $t\to 0^{+}$. Suppose ${\displaystyle \frac{1}{2}}<\delta <1$, and that f solves the homogeneous integral equation

$${\int }_0^{\infty }f\left(t\right)t^{\delta -1}{\displaystyle \frac{e^{xt}+\alpha }{e^{2xt}-1}}dt=0,x>0,$$

then $f=0$ almost everywhere on ${\mathbb{R}}_{+}$.

5. Conclusions

In this paper, we have investigated new aspects of the Mellin transform and its interplay with Lambert transforms. By developing a unified analytical framework, we derived inversion formulae that extend the applicability of Lambert transforms to a wider class of functions, including $L^p$ spaces with $1<p\le 2$. A key outcome of this study is the establishment of a Plancherel-type theorem, which enhances the utility of Lambert transforms in harmonic analysis and functional analysis settings.

The connection between the Mellin and Lambert transforms not only enables efficient inversion techniques but also provides a deeper insight into the structural properties of these integral transforms. Additionally, our results shed light on the uniqueness properties of the Lambert transforms, with further implications for Salem’s equivalence and its relation to the Riemann hypothesis. This underscores the relevance of Lambert transforms in addressing fundamental problems in analytic number theory.

The results obtained here open several directions for future research, particularly in the context of generalized functions and distribution theory, where the Mellin transform plays a central role. Subsequent investigations may explore extensions to other classes of integral transforms or their applications in spectral theory and the analysis of zeta functions.