Eñe Product and Its Ring Structure

• The eñe product is a binary operation on normalized formal power series that creates a second ring structure where series multiplication is additive and the eñe product is multiplicative.
• It acts by multiplying zeros pairwise in polynomials and extends to functions like rational, meromorphic, entire, and transalgebraic, with applications in divisor convolution.
• The operation is linked to Hadamard multiplication, logarithmic derivatives, and Witt vectors, offering insights into operator theory and connections to arithmetic and the Riemann Hypothesis.

The eñe product is a binary operation on normalized formal power series

$\Lambda(A):=1+zA[[z]]$

over a commutative ring $A$. It equips $\Lambda(A)$ with a second ring structure in which the usual multiplication of series is the additive law and the eñe product is the multiplicative law. On polynomials normalized by $f(0)=g(0)=1$, it acts by multiplying zeros pairwise; in exponential coordinates it becomes the diagonal operation

$F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$

for $F(z)=\sum_{n\ge0}A_nz^n$ and $G(z)=\sum_{n\ge0}B_nz^n$. The operation extends from formal series to rational, meromorphic, entire, and transalgebraic functions, is closely related to Hadamard multiplication and logarithmic derivatives, and, after inversion, coincides with the multiplication of the Big Witt ring (Pérez-Marco, 2019, Pérez-Marco, 2020, Pérez-Marco, 2019, Barsky et al., 31 May 2026).

1. Formal definition and ring-theoretic structure

For

$$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$

the eñe product is defined by

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$

where each coefficient $c_n$ is a universal polynomial with integer coefficients in

$A$0

This universality is essential: the definition makes sense over any commutative coefficient ring. The resulting structure $A$1 is a commutative ring in which usual multiplication of series is the additive law, the additive identity is the constant series $A$2, and the multiplicative identity for $A$3 is $A$4 (or $A$5 in the formal-variable notation of the algebraic papers) (Pérez-Marco, 2019).

The same operation is more transparent in exponential coordinates. If

$A$6

then

$A$7

where the exponential eñe product is

$A$8

Thus $A$9 is bilinear, commutative, and associative on logarithms, while $\Lambda(A)$0 is the induced operation on exponentials. In these coordinates the eñe product is diagonal in the monomial basis: there is no mixing of indices, only coefficientwise multiplication with weight $\Lambda(A)$1 (Pérez-Marco, 2019, Pérez-Marco, 2020).

Several structural identities follow immediately from the divisor interpretation and the universal formulas. For $\Lambda(A)$2,

$\Lambda(A)$3

and in particular

$\Lambda(A)$4

Moreover,

$\Lambda(A)$5

and if $\Lambda(A)$6,

$\Lambda(A)$7

These formulas show that $\Lambda(A)$8 is compatible with rescaling and with the multiplicative arithmetic of exponents (Pérez-Marco, 2019).

2. Multiplicative convolution of zeros and divisors

The defining geometric property of the eñe product is its action on zeros. If

$\Lambda(A)$9

with $f(0)=g(0)=1$0, then

$f(0)=g(0)=1$1

Hence the zeros of $f(0)=g(0)=1$2 are exactly the pairwise products $f(0)=g(0)=1$3. If $f(0)=g(0)=1$4 and $f(0)=g(0)=1$5 denote multiplicities, positive for zeros and negative for poles, then the multiplicity at $f(0)=g(0)=1$6 is

$f(0)=g(0)=1$7

In divisor notation,

$f(0)=g(0)=1$8

Equivalently, the eñe product on divisors is the bilinear extension of the rule

$f(0)=g(0)=1$9

This is the core sense in which the eñe product is a multiplicative convolution of divisors (Pérez-Marco, 2019, Pérez-Marco, 2020).

A more primitive formulation appears at the level of divisors on a semigroup $F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$0. For finite divisors

$F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$1

their convolution is

$F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$2

If $F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$3 is a semigroup, this defines an associative ring structure on finite divisors; if $F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$4 is commutative, the ring is commutative. Specializing to the multiplicative monoid $F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$5, and transporting the divisor convolution through the factorization

$F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$6

gives the eñe product on split rational functions normalized at $F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$7 (Barsky et al., 31 May 2026).

This divisor formalism extends beyond polynomials. For rational and meromorphic functions, the sign rules are: zero $F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$8 zero $F\star_e G(z)=-\sum_{n\ge1} n\,A_nB_n\,z^n$9 zero, pole $F(z)=\sum_{n\ge0}A_nz^n$0 pole $F(z)=\sum_{n\ge0}A_nz^n$1 zero, zero $F(z)=\sum_{n\ge0}A_nz^n$2 pole $F(z)=\sum_{n\ge0}A_nz^n$3 pole, and pole $F(z)=\sum_{n\ge0}A_nz^n$4 zero $F(z)=\sum_{n\ge0}A_nz^n$5 pole. In the rational case the eñe product leaves invariant

$F(z)=\sum_{n\ge0}A_nz^n$6

and degrees multiply: $F(z)=\sum_{n\ge0}A_nz^n$7 A plausible implication is that the divisor description, rather than the coefficient formulas, is the primary organizing principle of the theory (Pérez-Marco, 2019).

3. Exponential coordinates, Hadamard multiplication, and induced operators

The eñe product is tightly linked to the logarithmic derivative and to Hadamard multiplication. If

$F(z)=\sum_{n\ge0}A_nz^n$8

then

$F(z)=\sum_{n\ge0}A_nz^n$9

where $G(z)=\sum_{n\ge0}B_nz^n$0 denotes the Hadamard product

$G(z)=\sum_{n\ge0}B_nz^n$1

Equivalently, in exponential coordinates

$G(z)=\sum_{n\ge0}B_nz^n$2

so the eñe product is a Hadamard-type product twisted by the Koebe weight sequence $G(z)=\sum_{n\ge0}B_nz^n$3 (Pérez-Marco, 2019, Pérez-Marco, 2020).

This diagonalization yields a direct bridge to operator theory. For matrices $G(z)=\sum_{n\ge0}B_nz^n$4,

$G(z)=\sum_{n\ge0}B_nz^n$5

Thus tensor product of linear operators corresponds exactly to eñe multiplication of the associated characteristic-type polynomials. In particular, one may compute $G(z)=\sum_{n\ge0}B_nz^n$6 by realizing $G(z)=\sum_{n\ge0}B_nz^n$7 and $G(z)=\sum_{n\ge0}B_nz^n$8 as characteristic polynomials of companion matrices and passing to a tensor product (Pérez-Marco, 2019).

The same formalism produces Hecke-type operators. If

$G(z)=\sum_{n\ge0}B_nz^n$9

then

$$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$0

Thus eñe multiplication by $$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$1 extracts exactly those exponential coefficients whose indices are divisible by $$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$2. The associated Hecke operator is defined by

$$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$3

so that

$$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$4

Together with the dilatation operators $$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$5, these operators satisfy identities parallel to classical Hecke relations, including $$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$6 when $$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$7 (Pérez-Marco, 2019).

4. Relation to the Big Witt ring

The eñe product gives a natural construction of the Big Witt ring. Classically, the underlying set of the Big Witt ring $$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$8 is identified with

$$f(z)=1+\sum_{n\ge1} a_n z^n,\qquad g(z)=1+\sum_{n\ge1} b_n z^n,$$9

with addition given by usual multiplication of series and multiplication defined through ghost components. In the eñe approach, one starts instead from the elementary rule that zeros multiply, extends that rule continuously from split polynomials to all formal series by universal polynomial formulas, and only then recovers the Witt structure (Barsky et al., 31 May 2026).

The relevant exponential coordinates are Newton sums and ghost components. If

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$0

then

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$1

Under the eñe product,

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$2

so Newton sums multiply coordinatewise (Barsky et al., 31 May 2026).

For the Big Witt ring one uses instead

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$3

where

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$4

are the Bergman–Witt ghost polynomials. The central result is that the twisted eñe product

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$5

coincides with the Big Witt product $f\star g(z)=1+c_1z+c_2z^2+\cdots,$6. Equivalently, after inversion the eñe ring is exactly the classical Big Witt ring (Barsky et al., 31 May 2026).

The construction is also functorial. A ring morphism $f\star g(z)=1+c_1z+c_2z^2+\cdots,$7 induces a coefficientwise map

$f\star g(z)=1+c_1z+c_2z^2+\cdots,$8

that preserves the eñe product. This places the eñe construction and the Big Witt functor in the same categorical framework (Barsky et al., 31 May 2026).

5. Singularities, monodromy, and invariant analytic classes

The analytic theory of the eñe product is organized by monodromy. For a holomorphic function $f\star g(z)=1+c_1z+c_2z^2+\cdots,$9 on a punctured neighborhood of $c_n$0, let

$c_n$1

be the monodromy after one positive turn around $c_n$2. Holomorphic monodromy means that $c_n$3 extends holomorphically to $c_n$4. In that case,

$c_n$5

with $c_n$6 having a uniform singularity at $c_n$7. Integrable singularities are exactly those with totally holomorphic monodromy (Pérez-Marco, 2020).

The Hadamard product and the exponential eñe product admit parallel convolution formulas: $c_n$8 and

$c_n$9

The crucial difference is the absence of the explicit $A$00 factor in the $A$01 kernel. From this one obtains explicit monodromy formulas at product singularities $A$02. If $A$03 and $A$04 have isolated singularities with holomorphic monodromy at $A$05 and $A$06, then the singularities of $A$07 and of $A$08 are again contained in the set of products $A$09, are isolated, and have holomorphic monodromy. In the totally holomorphic case, the $A$10 monodromy formula contains no $A$11 singular kernel, so the origin does not become a new ramification point (Pérez-Marco, 2020).

These formulas imply closure properties for natural analytic classes. If $A$12 contains $A$13, the class $A$14 consists of germs holomorphic at $A$15 whose singularities lie in $A$16 and whose local monodromies belong to $A$17. The ring $A$18 is closed under Hadamard product and under the exponential eñe product $A$19; moreover it is the minimal Hadamard, respectively eñe, ring containing functions with polynomial monodromies in $A$20 and singularities in $A$21 (Pérez-Marco, 2020).

The same analytic control extends to entire functions. If $A$22 denotes the set of entire functions of order $A$23 with constant term $A$24, then $A$25 is stable under $A$26, and the product is continuous. For entire functions of finite genus $A$27, the eñe product respects Hadamard–Weierstrass factorization: $A$28 implies

$A$29

At the level of convergence, the eñe-radius satisfies

$A$30

and $A$31 is continuous for the corresponding compact-open topology (Pérez-Marco, 2019).

6. Transalgebraic extension, polylogarithms, and arithmetic context

The eñe product extends beyond meromorphic functions to the transalgebraic class. A transalgebraic function on a compact Riemann surface is meromorphic outside finitely many points and has only finite-order exponential singularities at the punctures. Locally such a singularity has the form

$A$32

with $A$33 meromorphic; the order $A$34 of the exponential singularity is finite exactly when $A$35 has a pole of order $A$36. On the Riemann sphere, every transalgebraic function is of the form

$A$37

with $A$38 and $A$39. Modulo nonzero constants, the transalgebraic class becomes a commutative graded topological ring in which usual multiplication is the additive structure and the eñe product is the multiplicative structure (Pérez-Marco, 2019).

A distinguished hierarchy is generated by Euler’s rational functions

$A$40

with recurrence

$A$41

They encode higher-order “infinite zeros” through

$A$42

Their eñe multiplication is rigid: $A$43 Thus the support multiplies and the order adds, exactly as in divisor convolution (Pérez-Marco, 2019).

Polylogarithms provide the dual hierarchy. For

$A$44

one has

$A$45

Their monodromy at $A$46 is

$A$47

In the transalgebraic framework, the negative-index continuation of the $A$48-hierarchy satisfies

$A$49

and

$A$50

This motivates the interpretation of $A$51 as an eñe-pole of order $A$52 (Pérez-Marco, 2020, Pérez-Marco, 2019).

The arithmetic motivation of the theory comes from Euler products, zeta functions, and Dirichlet $A$53-functions. In that setting, local factors may be written in normalized form $A$54, and the eñe product models multiplicative interaction of their zero data. The papers emphasize that the eñe product plays a central role in work on “statistics on Riemann zeros” and on heuristic aspects of the Riemann Hypothesis. This suggests a broader interpretation of the eñe ring as a framework in which divisor convolution, Hadamard multiplication, Witt vectors, and analytic continuation are different manifestations of the same underlying multiplicative geometry of zeros (Pérez-Marco, 2019).