Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis

1. Introduction and Preliminaries

A Lambert series refers to any series expressed in the form of

$\begin{matrix} \sum_{k = 1}^{\infty} α_{k} \frac{x^{k}}{1 - x^{k}}, | x | < 1, α_{k} \in C . \end{matrix}$

(1)

It represents a natural extension of the following number theory formulae, which have been demonstrated by Lambert [1]:

$\begin{matrix} \sum_{k = 1}^{\infty} \frac{x^{k}}{1 - x^{k}} = \sum_{k = 1}^{\infty} τ (k) x^{k}, \sum_{k = 1}^{\infty} k \frac{x^{k}}{1 - x^{k}} = \sum_{k = 1}^{\infty} σ (k), | x | < 1, \end{matrix}$

where $σ (k)$ denotes the sum of the divisors of k, while $τ (k)$ denotes number of the divisors of k, respectively. Widder [2] introduced the Lambert transform as an extension of the Lambert series (1).

Various researchers explored different aspects of the Lambert transform and its inversion formulae. Noteworthy contributions include those by Miller [3], who investigated the convergence properties essential for developing inversion formulae for the Lambert transform and introduced summability techniques for power series utilizing the Lambert transform [4]. Ferreira and López [5] obtained asymptotic expansions of the Lambert transform for large and small values of the variable. Widder [2] derived an inversion formula involving the Möbius function as a limit of derivatives. Several inversion formulae related to Widder’s formula have been examined in works such as [6,7,8]. Raina and Nahar [9] extended the Lambert transform to a class of functions connected to the Hurwitz zeta function. Raina and Srivastava [10] introduced a generalized Lambert transform associated with the generalized Riemann zeta function. Goyal and Laddha [11] introduced generalizations of the Riemann zeta function and the generalized Lambert transform. An inversion formula for the generalized Lambert transform is provided. Goldberg [12] proposed a more generalized kernel for the Lambert transform, working with transforms as Stieltjes integrals, and derived inversion formulae. Furthermore, Hayek et al. [13] investigated this generalized Lambert transform on the space of distributions of compact support, providing an inversion formula applicable to this space. Recently, Maan et al. [14] extended this transform to Boehmian spaces.

This paper is divided into five sections. Section 1 serves as an introduction, providing basic definitions and notations utilized throughout this paper. In Section 2, we establish an interesting connection between the Mellin transform (2) and the Widder–Lambert transform (3) for some specific class of functions. In Section 3, motivated by the connection established in Section 2, we obtain a Plancherel theorem for the Widder–Lambert transform (3) (via the Mellin transform). As a consequence of this result, we prove that if f is a bounded measurable function such that

f (t) = O (t^{1 / 2})

t \to 0^{+}

, and

L_{δ} [f] = 0

\frac{1}{2} < δ < 1

, then

f = 0

almost everywhere on

R_{+}

. This is a significant approach to the Salem criterion and thus to the Riemann hypothesis. Section 4 presents an inversion formula for the Widder–Lambert transform (3) over Lebesgue integrable functions. In Section 5, an inversion formula for the Widder–Lambert transform is provided for distributions of compact support. Finally, Section 6 offers an approach to the Salem equivalence to the Riemann hypothesis via regular distributions of compact support. Finally, Section 7 provides concluding notes.

The Riemann hypothesis stands as a pivotal challenge within the realm of mathematics, inspiring numerous related hypotheses and conceptual developments. Notably, in 1953, the Greek mathematician Raphaël Salem established the boundedness of a certain integral involving Fourier coefficients that was equivalent to the truth of the Riemann hypothesis, as documented in [15]. This condition, proven to be both necessary and sufficient, serves as an equivalence to the Riemann hypothesis. Consequently, the problem can be forwarded in the setting of the integral transforms.

The Riemann zeta function

ζ (s)

, as referenced in [16], is expressed through the convergence of the series:

$\begin{matrix} ζ (s) = \sum_{k = 1}^{\infty} \frac{1}{k^{s}}, ℜ (s) > 1 . \end{matrix}$

Additionally, the Dirichlet zeta function $η (s)$ is denoted by the convergent series:

$\begin{matrix} η (s) = \sum_{k = 1}^{\infty} \frac{{(- 1)}^{k - 1}}{k^{s}}, ℜ (s) > 0 . \end{matrix}$

The series obtained by combining the Riemann zeta function $ζ (s)$ , and the Dirichlet zeta function $η (s)$ is given by

$\begin{matrix} η (s) = (1 - 2^{1 - s}) ζ (s), ℜ (s) > 1, \end{matrix}$

which extends $ζ (s)$ to $0 < ℜ (s) < 1 .$

It is a widely acknowledged fact that $ζ (s)$ (and consequently $η (s)$ ) exhibits no zeros for $ℜ (s) > 1$ .

The Salem equivalence to the Riemann hypothesis (Salem [15] and Broughan [17] (§ 8.4, pp. 139–142)) guarantees the absence of zeros of the Riemann zeta function within the strip

0 < ℜ (s) < 1

, where

ℜ (s) \neq \frac{1}{2}

The Mellin transform of a suitable function f is given as [18]

$(M [f]) (s) = \int_{0}^{\infty} t^{s - 1} f (t) d t, s \in C .$

(2)

In this paper, motivated by the integral Equation (1), we consider the Widder–Lambert integral transform (cf. [10] (Equation (4.1)) and [19] (Equation (2.15))) for each

δ > 0

as follows:

$(L_{δ} [f]) (x) = \int_{0}^{\infty} \frac{t^{δ - 1}}{e^{x t} + 1} f (t) d t, x > 0,$

(3)

where f is a suitable complex-valued function such that the integral converges.

We also denote, by

L_{a, p} (R_{+})

a \in R

, and

1 \leq p < \infty

, the Banach space of the complex-valued functions f on

R_{+}

with the norm

${∥ f ∥}_{a, p} = {(\int_{0}^{\infty} {|f (t)|}^{p} t^{a p - 1} d t)}^{1 / p} .$

Let K be a compact subset of $R_{+}$ . $D_{K} (R_{+})$ is the set of all complex-valued smooth functions defined on $R_{+}$ which vanish on those points on $R_{+}$ that are not in K.

For each non-negative integer n, we define the seminorm

$\begin{matrix} γ_{n} (ϕ) = max_{t \in R_{+}} | D_{t}^{n} ϕ (t) |, ϕ \in D_{K} (R_{+}), \end{matrix}$

$D_{t}^{n}$ denotes the n-th derivative with respect to variable t. With this family of seminorms, this space becomes a Fréchet space.

Now,

D (R_{+})

is the countable-union space of the spaces

D_{K} (R_{+})

[20] (§1.7, pp. 14–17).

D^{'} (R_{+})

denotes the dual of

D (R_{+})

[20] (§1.9, pp. 24–25), and it is called the space of distributions on

R_{+}

[20] (§2.2, pp. 32–36).

The space

E (R_{+})

is defined as the vector space of all infinitely differentiable complex-valued functions defined in

R_{+}

. Let

γ_{K, n}

be the seminorm on

E (R_{+})

defined by

$\begin{matrix} γ_{K, n} (ϕ) = max_{t \in K} | D_{t}^{n} ϕ (t) |, ϕ \in E (R_{+}), \end{matrix}$

for all $n \in N \cup 0$ , all compact set $K \subset R_{+}$ , and $D_{t}^{n}$ denotes the n-th derivative with respect to variable t. With this family of seminorms, this space becomes a Fréchet space. As usual, we denote the space $E^{'} (R_{+})$ as the dual space of $E (R_{+})$ . This $E^{'} (R_{+})$ agrees with the space of distributions on $R_{+}$ of compact support contained in $R_{+}$ [20] (§2.3, pp. 36–38).

3. Plancherel Theorem and the Salem Equivalence to the Riemann Hypothesis

By means of the connection established in the previous section between the Mellin transform and the Widder–Lambert transform, we obtain a Plancherel theorem for the Widder–Lambert transform (3). As it has been exposed in [21] (p. 694), the Mellin transform is defined for

f \in L_{a, 2} (R_{+})

by the integral

$(M [f]) (s) = \int_{0}^{\infty} t^{s - 1} f (t) d t, (Re s = a)$

(6)

being convergent in mean with respect to the norm in $L_{2} ((a - i \infty, a + i \infty))$ .

Also, for the case when

f \in L_{a, 1} (R_{+}) \cap L_{a, 2} (R_{+})

, the Mellin transform (6) agrees almost everywhere with the usual Mellin transform (2) (see [18] for details).

Thus, according to the Plancherel theorem for the Mellin transform given by Titchmarsh [18] (Theorem 71, pp. 94–95) and using Theorem 1 above, one has

Theorem 2 (Plancherel theorem).

Set

δ > 0

. Assume that f is a measurable function on

R_{+}

, such that the integral

$(L_{δ} [f]) (x) = \int_{0}^{\infty} f (t) \frac{t^{δ - 1}}{e^{x t} + 1} d t, x > 0,$

converges. Moreover, let $f \in L_{a, 1} (R_{+}) \cap L_{a, 2} (R_{+})$ , for some arbitrary a with $a < δ - 1$ . Then,

$\begin{matrix} \int_{0}^{\infty} {| f (x) |}^{2} x^{2 a - 1} d x = \frac{1}{2 π} \int_{- \infty}^{\infty} {|\frac{(M [L_{δ} [f]]) (δ - a - i t)}{Γ (δ - a - i t) η (δ - a - i t)}|}^{2} d t, \end{matrix}$

(7)

where $M$ denotes the Mellin transform (2) and $L_{δ}$ denotes the Widder–Lambert integral transform (3).

Proof.

In fact, since

f \in L_{a, 2} (R_{+})

, one obtains the following from [18] (Theorem 71, pp. 94–95):

$\begin{matrix} \int_{0}^{\infty} {| f (x) |}^{2} x^{2 a - 1} d x = \frac{1}{2 π} \int_{- \infty}^{\infty} {|(M [f]) (a + i t)|}^{2} d t, \end{matrix}$

where $M$ denotes the $L_{a, 2}$ Mellin transform (6).

Now, since

f \in L_{a, 1} (R_{+})

, the Mellin transform (6) agrees almost everywhere on

R_{+}

with the usual Mellin transform (2). Thus, using Theorem 1, one obtains the relation (7). □

Corollary 1.

Assuming the hypothesis of Theorem 2, one has

$I f L_{δ} [f] = 0 a l m o s t e v e r y w h e r e o n R_{+} t h e n f = 0 a l m o s t e v e r y w h e r e o n R_{+} .$

Proof.

Since

L_{δ} [f] = 0

almost everywhere on

R_{+}

, the right-hand side of (7) is zero. Thus,

$\begin{matrix} \int_{0}^{\infty} {| f (x) |}^{2} x^{2 a - 1} d x = 0, \end{matrix}$

and so $f = 0$ almost everywhere on $R_{+}$ . □

Now, concerning the Salem equivalence of the Riemann hypothesis and taking into account that, for any bounded measurable function, the integral in (3) converges, one obtains the following:

Corollary 2.

Let f be a bounded measurable function such that

f (t) = O (t^{1 / 2})

t \to 0^{+}

. Set

\frac{1}{2} < δ < 1

and assume that f is a solution of the homogeneous equation

$(L_{δ} [f]) (x) = \int_{0}^{\infty} \frac{t^{δ - 1}}{e^{x t} + 1} f (t) d t, x > 0 .$

Then, $f = 0$ almost everywhere on $R_{+}$ .

Proof.

Observe that, for $\frac{1}{2} < δ < 1$ , one obtains $\frac{1}{2} = 1 - \frac{1}{2} > 1 - δ$ . Taking a such that $\frac{1}{2} > - a > 1 - δ$ , one has that the class of functions of this corollary satisfies the hypothesis of the theorem above. Then, the result holds. □

Remark 1.

The result obtained in Corollary 2 agrees with the result obtained in Corollary 3.3 of [16]. However, in contrast to the proof of Corollary 3.3 of [16] in the present proof, we do not use any inversion formulae, such as Post–Widder or

L_{2}

inversion formulae. Instead, we made use of a Plancherel theorem. This proof represents the novelty of our work.

5. An Inversion Formula for the Widder–Lambert Transform on $E^{'} (R_{+})$

In this section, we study the Widder–Lambert transform over distributions of compact support on $R_{+}$ . Here, we obtain an inversion formula for this transform and we consider the injectivity of this distributional transform.

Definition 1.

Set

δ > 0

. The Widder–Lambert transform on

E^{'} (R_{+})

is defined by

$\begin{matrix} (L_{δ} [f]) (x) = 〈f (t), \frac{t^{δ - 1}}{e^{x t} + 1}〉, x > 0, f \in E^{'} (R_{+}) . \end{matrix}$

(9)

From the following relation:

$\begin{matrix} \frac{1}{e^{x} + 1} = \frac{1}{e^{x} - 1} - \frac{2}{e^{2 x} - 1}, \end{matrix}$

valid for any $x \in R_{+}$ , one has

$\begin{matrix} (L_{δ} [f]) (x) = 〈f (t), \frac{t^{δ - 1}}{e^{x t} + 1}〉 & = & 〈f (t), \frac{t^{δ - 1}}{e^{x t} - 1}〉 - 〈f (t), \frac{2 t^{δ - 1}}{e^{2 x t} - 1}〉 \\ = & 〈f (t), \frac{t^{δ - 1}}{e^{x t} - 1}〉 - 〈f (\frac{t}{2}), \frac{2^{1 - δ} t^{δ - 1}}{e^{x t} - 1}〉 \\ = & 〈f (t) t^{δ - 1} - f (\frac{t}{2}) 2^{1 - δ} t^{δ - 1}, \frac{1}{e^{x t} - 1}〉, x > 0, \end{matrix}$

which is the distributional Lambert transform ([13], (2.3) when $a_{k} = 1, for all k \in N$ ) of the member $[f (t) - f (\frac{t}{2}) 2^{1 - δ}] t^{δ - 1}$ of $E^{'} (R_{+})$ .

From [13] (Proposition 2.1), the distributional Lambert transform of

f \in E^{'} (R_{+})

is a smooth function. Then, the Widder–Lambert transform (9) of f becomes a member of

E (R_{+})

Now, using the inversion formula ([13], Theorem 2.1) of the distributional Lambert transform over

E^{'} (R_{+})

, one has

$\begin{matrix} lim_{p \to \infty} \frac{{(- 1)}^{p}}{p!} {(\frac{p}{t})}^{p + 1} \sum_{n = 1}^{\infty} μ (n) n^{p} {(L_{δ} [f])}^{(p)} (\frac{n p}{t}) = [f (t) - f (\frac{t}{2}) 2^{1 - δ}] t^{δ - 1}, \end{matrix}$

in the sense of the convergence in $D^{'} (R_{+})$ .

Thus, one has

$\begin{matrix} - \sum_{k = 1}^{\infty} [lim_{p \to \infty} \frac{{(- 1)}^{p}}{p!} {(\frac{p}{2^{k} t})}^{p + 1} \sum_{n = 1}^{\infty} μ (n) n^{p} {(L_{δ} [f])}^{(p)} (\frac{n p}{2^{k} t})] \\ = - \sum_{k = 1}^{\infty} [f (2^{k} t) - f (2^{k - 1} t) 2^{1 - δ}] 2^{k (δ - 1)} t^{δ - 1} \\ = - lim_{N \to \infty} \sum_{k = 1}^{N} [f (2^{k} t) - f (2^{k - 1} t) 2^{1 - δ}] 2^{k (δ - 1)} t^{δ - 1} \\ = lim_{N \to \infty} [f (t) t^{δ - 1} - f (2^{N} t) 2^{N (δ - 1)} t^{δ - 1}], \end{matrix}$

in the sense of the convergence in $D^{'} (R_{+})$ .

Observe that

$\begin{matrix} lim_{N \to \infty} 〈{(2^{N} t)}^{δ - 1} f (2^{N} t), ϕ (t)〉 = 0, for all ϕ \in D (R_{+}) . \end{matrix}$

In fact, for any $ϕ \in D (R_{+})$ , we have

$\begin{matrix} lim_{N \to \infty} 〈{(2^{N} t)}^{δ - 1} f (2^{N} t), ϕ (t)〉 \\ = lim_{N \to \infty} 〈t^{δ - 1} f (t), \frac{ϕ (\frac{t}{2^{N}})}{2^{N}}〉 . \end{matrix}$

Now, $\frac{ϕ (\frac{t}{2^{N}})}{2^{N}} \to 0$ as $N \to \infty$ in $E (R_{+})$ .

In fact, for any compact

K \subseteq R_{+}

m \in N \cup 0

, one has

$\begin{matrix} max_{t \in K} |\frac{D^{m} ϕ (\frac{t}{2^{N}})}{2^{N}}| \leq \frac{M_{m}}{2^{N}}, \end{matrix}$

with it being $ϕ \in D (R_{+})$ . Thus, $s u p p ϕ \subseteq [a, b] \subseteq R_{+}$ , $M_{m} = {max}_{t \in [a, b]} | D^{m} ϕ (t) |$ .

Therefore,

\frac{ϕ (\frac{t}{2^{N}})}{2^{N}} \to 0

N \to \infty

E (R_{+})

, and with it being

t^{δ - 1} f (t) \in E^{'} (R_{+})

, we have that

$\begin{matrix} lim_{N \to \infty} 〈t^{δ - 1} f (t), \frac{ϕ (\frac{t}{2^{N}})}{2^{N}}〉 \to 0 a s N \to \infty . \end{matrix}$

Thus,

$\begin{matrix} lim_{N \to \infty} 〈{(2^{N} t)}^{δ - 1} f (2^{N} t), ϕ (t)〉 = 0, for all ϕ \in D (R_{+}), \end{matrix}$

and so we have proven the following results:

Theorem 4.

Set

δ > 0

. For

f \in E^{'} (R_{+})

and

$(L_{δ} [f]) (x) = 〈f (t), \frac{t^{δ - 1}}{e^{x t} + 1}〉, x > 0,$

we have

(10)

in the sense of the convergence in $D^{'} (R_{+})$ (weak convergence) [20] (§1.9, p. 24). Here, $μ (n)$ numbers are the Möbius numbers defined as $μ = 1, μ_{n} = {(- 1)}^{s}$ if n is the product of s distinct primes, and $μ_{n} = 0$ if n is divisible by a square and ${(L_{δ} [f])}^{(p)} (z)$ denotes the conventional p-th derivative of $(L_{δ} [f]) (z)$ with respect to its argument z.

Corollary 5 (Inversion formula).

Assuming the hypothesis of Theorem 4, one has

(11)

in the sense of the convergence in $D^{'} (R_{+})$ .

As an immediate consequence of Corollary 5, one has

Corollary 6 (Injectivity).

Set

δ > 0

. The map

L_{δ} : E^{'} (R_{+}) \to E (R_{+})

defined by

$(L_{δ} [f]) (x) = 〈f (t), \frac{t^{δ - 1}}{e^{x t} + 1}〉, x > 0, f \in E^{'} (R_{+}),$

is one-to-one.

Proof.

From (11), if

(L_{δ} [f]) = 0

, then

〈t^{δ - 1} f, ϕ〉 = 0

for all

ϕ \in D (R_{+})

and so

〈f, ϕ〉 = 0

for all

ϕ \in D (R_{+})

. Now, since

D (R_{+})

is dense in

E (R_{+})

[20] (§2.3, p. 37) and with it being

f \in E^{'} (R_{+})

, then

〈f, ϕ〉 = 0

for all

ϕ \in E (R_{+})

. Thus,

f = 0

E^{'} (R_{+})

. □

6. Regular Distributions of Compact Support Versus the Salem Equivalence to the Riemann Hypothesis

In this section, we consider the distributional Widder–Lambert transform (9) over regular distributions of compact support on

R_{+}

. From this, one arrives at an interesting connection with the Salem equivalence to the Riemann hypothesis.

For the case when f is a bounded measurable function with compact support in

R_{+}

, the functional

T_{f}

over

E (R_{+})

given by

$\begin{matrix} 〈T_{f}, ϕ〉 = \int_{0}^{\infty} f (x) ϕ (x) d x, ϕ \in E (R_{+}), \end{matrix}$

becomes a member in $E^{'} (R_{+})$ , which is called a regular member.

Set

δ > 0

. Observe that by taking the following for each

x > 0

$\begin{matrix} ϕ (t) = \frac{t^{δ - 1}}{e^{x t} + 1}, t > 0, \end{matrix}$

one has

$\begin{matrix} 〈T_{f}, \frac{t^{δ - 1}}{e^{x t} + 1}〉 = \int_{0}^{\infty} f (t) \frac{t^{δ - 1}}{e^{x t} + 1} d t, \end{matrix}$

i.e., the distributional Widder–Lambert transform (9) of $T_{f}$ agrees with the classical Widder–Lambert transform (3) of the function f.

Concerning the Salem equivalence to the Riemann hypothesis, this equivalence is given by the following:

For

\frac{1}{2} < δ < 1

, any bounded measurable function f on

R_{+}

satisfying the integral equation

$\begin{matrix} \int_{0}^{\infty} f (t) \frac{t^{δ - 1}}{e^{x t} + 1} d t = 0, for all x > 0, \end{matrix}$

is zero almost everywhere on $R_{+}$ ([15,16,17,19,22] and (§ 8.4, pp. 139–142)).

In our present setting, we prove the following result:

Proposition 1.

For

\frac{1}{2} < δ < 1

, any bounded measurable function f on

R_{+}

with compact support contained in

R_{+}

satisfying the integral equation

$\begin{matrix} \int_{0}^{\infty} f (t) \frac{t^{δ - 1}}{e^{x t} + 1} d t = 0, f o r a l l x > 0, \end{matrix}$

is zero almost everywhere on $R_{+}$ .

Proof.

Observe that

$\begin{matrix} 〈T_{f}, \frac{t^{δ - 1}}{e^{x t} + 1}〉 = \int_{0}^{\infty} f (t) \frac{t^{δ - 1}}{e^{x t} + 1} d t = 0, for all x > 0 . \end{matrix}$

Thus, using Corollary 6, one has that $T_{f} = 0$ . Now, from [23] (pp. 8–9), it follows that $f = 0$ almost everywhere on $R_{+}$ . Therefore, the proposition holds. □

Remark 3.

The result obtained in Proposition 1 is a particular case of the result obtained in Corollary 2. However, Proposition 1 sheds light on how to use the theory of distributions for obtaining a class of functions satisfying the Salem equivalence to the Riemann hypothesis.

If one could obtain an inversion formula of the Widder–Lambert transform (9) to a broad space of distributions (and not only to distributions of compact support), one would obtain a larger class of functions that satisfies the Salem equivalence to the Riemann hypothesis.

7. Final Observations and Conclusions

In conclusion, this paper systematically investigates Plancherel’s theorem and the inversion formulae for the Widder–Lambert transform across various mathematical frameworks, including Lebesgue integrable functions, distributions with compact support, and regular distributions of compact support. A key contribution lies in leveraging the Plancherel theorem for the Mellin transform alongside the connection between the Mellin and Widder–Lambert transforms. Notably, this approach departs from traditional inversion formulae such as the Post–Widder or

L_{2}

inversion formulae. Instead, it relies on Plancherel’s theorem, as detailed in Theorem 71 of Titchmarsh’s work [18] (Theorem 71, pp. 94–95). This innovative methodology underscores the novelty of our research.

In summary, the validity of the Riemann hypothesis could be supported by proving Corollary 2 for bounded measurable functions without other restrictions. However, this paper shows that bounded measurable functions f which satisfy $f (t) = O (t^{1 / 2})$ as $t \to 0^{+}$ meet Salem’s criterion. Therefore, any bounded measurable functions f that are a counterexample to Salem’s equivalence (and thus to Riemann hypothesis) must not have the property $f (t) = O (t^{1 / 2})$ as $t \to 0^{+}$ .

These findings not only reinforce the link between Salem’s equivalence and the Riemann hypothesis but also open new avenues for exploring this hypothesis through the lens of integral transforms.

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Emilio R. Negrín www.mdpi.com

Greenberg News

Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis

1. Introduction and Preliminaries

3. Plancherel Theorem and the Salem Equivalence to the Riemann Hypothesis

5. An Inversion Formula for the Widder–Lambert Transform on $E^{'} (R_{+})$

6. Regular Distributions of Compact Support Versus the Salem Equivalence to the Riemann Hypothesis

7. Final Observations and Conclusions

Greenberg

1. Introduction and Preliminaries

3. Plancherel Theorem and the Salem Equivalence to the Riemann Hypothesis

5. An Inversion Formula for the Widder–Lambert Transform on E ′ R +

6. Regular Distributions of Compact Support Versus the Salem Equivalence to the Riemann Hypothesis

7. Final Observations and Conclusions

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5. An Inversion Formula for the Widder–Lambert Transform on $E^{'} (R_{+})$