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Ricardo Pérez-Marco

Publications and source records attributed to Ricardo Pérez-Marco.

At least 19 recordsLinked to original sources
arXiv2026

The eñe product over a commutative ring

We define the eñe product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the additive structure and the eñe product is the multiplicative one. For polynomials with complex coefficients, the eñe product acts as a multiplicative convolution of their divisor. We study its algebraic properties, its relation to symmetric functions on an infinite number of variables, to tensor products, and Hecke operators. The exponential linearizes also the eñe product. The eñe product extends to rational functions and formal meromorphic functions. We also study the analytic properties over the complex numbers, and for entire functions. The eñe product respects Hadamard-Weierstrass factorization and is related to the Hadamard product. The eñe product plays a central role in predicting the phenomenon of the "statistics on Riemann zeros" for Riemann zeta function and general Dirichlet $L$-functions discovered by the author. It also gives reasons to believe in the Riemann Hypothesis as explained in the survey "Notes on the Riemann Hypothesis".

math.CA↗
arXiv2025

The Cantor Riemannium

The Riemann surface of a holomorphic germ is the space generated by its Weierstrass analytic continuation. The Riemannium space of a holomorphic germ is the space generated by its Borel monogenic continuation. Riemannium spaces are metric, path connected, Gromov length spaces, not necessarily $σ$-compact. We construct an example of Riemannium space: The Cantor Riemannium.

math.CV↗
arXiv2025

Monodromies of singularities of the Hadamard and eñe product

We prove that singularities with holomorphic monodromies are preserved by the Hadamard product. We find an explicit formula for the monodromy of the singularities, and similar formulas for the exponential eñe product. Using these formulas we get new direct proofs of classical results and the invariance of some rings of functions by Hadamard and eñe product (which is the product of the big Witt ring). This explains the good behavior of the eñe product on divisors.

math.CV↗
arXiv2023

On the genesis of BBP formulas

We present a general procedure to generate infinitely many BBP and BBP-like formulas for the simplest transcendental numbers. This provides some insight and a better understanding into their nature. In particular, we can derive the main known BBP formulas for $π$. We can understand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas representing $0$ come from. We also explain what is the observed relation between some BBP formulas for $\log 2$ and $π$, that are obtained by taking real and imaginary parts of a general complex BBP formula. Our methods are elementary, but motivated by transalgebraic considerations, and offer a new way to obtain and to search many new BBP formulas and, conjecturally, to better understand transalgebraic relations between transcendental constants.

math.NT↗
arXiv2022

On the definition of higher Gamma functions

We generalize our previous new definition of Euler Gamma function to higher Gamma functions. With this unified approach, we characterize Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson $q$-Gamma function, and Nishizawa higher $q$-Gamma functions. This approach extends to more general functional equations. This generalization reveals the multiplicative group structure of solutions of the functional equation that appears as a cocycle equation. We also generalize Barnes hierarchy of higher Gamma function and multiple Gamma functions. In this new approach, Barnes-Hurwitz zeta functions are no longer required for the definition of Barnes multiple Gamma functions. This simplifies the classical definition, without the necessary analytic preliminaries about the meromorphic extension of Barnes-Hurwitz zeta functions, and defines a larger class of Gamma functions. For some algebraic independence conditions on the parameters, we have uniqueness of the solutions, which implies the coincidence of our multiple Gamma functions with Barnes multiple Gamma functions.

math.CV↗
arXiv2021

On the definition of Euler Gamma function

We present a new definition of Euler Gamma function. From the complex analysis and transalgebraic viewpoint, it is a natural characterization in the space of finite order meromorphic functions. We show how the classical theory and formulas develop naturally, and we discuss the relation with other definitions. We show in a companion article, that this definition generalizes to higher gamma functions and provides an unifying framework for their definition which is more natural than the usual Bohr-Mollerup or Lerch approaches.

math.CV↗
arXiv2020

Notes on the historical bibliography of the gamma function

Telegraphic notes on the historical bibliography of the Gamma function and Eulerian integrals. Correction to some classical references. Some topics of the interest of the author. We provide some extensive (but not exhaustive) bibliography. Feedback is welcome, notes will be updated and some references need completion.

math.HO↗
arXiv2020

Local monodromy formula of Hadamard products

We find an explicit general formula for the iterated local monodromy of singularities of the Hadamard product of functions with integrable singularities. The formula implies the invariance by Hadamard product of the class of functions with integrable singularities with recurrent monodromies. In particular, it implies the recurrence of the local monodromy of functions with finite Hadamard grade as defined by Allouche and Mendès-France. We give other examples of natural classes of functions with recurrent monodromies, functions with algebro-logarithmic singularities, and more generally with polylogarithm monodromies. We sketch applications to elliptic integrals, hypergeometric functions, and to fractional integration.

math.CV↗
arXiv2020

Profit lag and alternate network mining

For a mining strategy we define the notion of "profit lag" as the minimum time it takes to be profitable after that moment. We compute closed forms for the profit lag and the revenue ratio for the strategies "selfish mining" and "intermittent selfish mining". This confirms some earlier numerical simulations and clarifies misunderstandings on profitability in the literature. We also study mining pairs of PoW cryptocurrencies, often coming from a fork, with the same mining algorithm. This represents a vector of attack that can be exploited using the "alternate network mining" strategy that we define. We compute closed forms for the profit lag and the revenue ratiofor this strategy that is more profitable than selfish mining and intermittent selfish mining. It is also harder to counter since it does not rely on a flaw in the difficulty adjustment formula that is the reason for profitability of the other strategies.

cs.CR↗
arXiv2020

Double spend races

We correct the double spend race analysis given in Nakamoto's foundational Bitcoin article and give a closed-form formula for the probability of success of a double spend attack using the Regularized Incomplete Beta Function. We give a proof of the exponential decay on the number of confirmations, often cited in the literature, and find an asymptotic formula. Larger number of confirmations are necessary compared to those given by Nakamoto. We also compute the probability conditional to the known validation time of the blocks. This provides a finer risk analysis than the classical one.

cs.CR↗
arXiv2020

The mathematics of Bitcoin

We survey recent results on the mathematical stability of Bitcoin protocol. Profitability and probability of a double spend are estimated in closed form with classical special functions. The stability of Bitcoin mining rules is analyzed and several theorems are proved using martingale and combinatorics techniques. In particular, the empirical observation of the stability of the Bitcoin protocol is proved. This survey article on the mathematics of Bitcoin is published by the Newsletter of the European Mathematical Society, vol.115, 2020, p.31-37. Continuation of arXiv:1601.05254 (EMS Newsletter, 100, 2016 p.32).

math.HO↗
arXiv2020

Ant Routing scalability for the Lightning Network

The ambition of the Lightning Network is to provide a second layer to the Bitcoin network to enable transactions confirmed instantly, securely and anonymously with a world scale capacity using a decentralized protocol. Some of the current propositions and implementations present some difficulties in anonymity, scaling and decentalization. The Ant Routing algorithm for the Lightning Network was proposed in \cite{GrunspanPerez} for maximal decentralization, anonymity and potential scaling. It solves several problems of current implementation, such as channel information update and centralization by beacon nodes. Ant Routing nodes play all the same role and don't require any extra information on the network topology beside for their immediate neighbors. The goal of LN transactions are completed instantaneously and anonymously. We study the scaling of the Ant Routing protocol. We propose a precise implementation, with efficient memory management using AVL trees. We evaluate the efficiency of the algorithm and we estimate the memory usage of nodes by local node workload simulations. We prove that the number of transactions per second that Ant Routing can sustain is of the order of several thousands which is enough for a global payment network.

cs.DC↗
arXiv2019

Eñe product in the transalgebraic class

We define transalgebraic functions on a compact Riemann surface as meromorphic functions except at a finite number of punctures where they have finite order exponential singularities. This transalgebraic class is a topological multiplicative group. We extend the action of the eñe product to the transcendental class on the Riemann sphere. This transalgebraic class, modulo constant functions, is a commutative ring for the multiplication, as the additive structure, and the eñe product, as the multiplicative structure. In particular, the divisor action of the eñe product by multiplicative convolution extends to these transalgebraic divisors. The polylogarithm hierarchy appears related to transalgebraic eñe poles of higher order.

math.CV↗
arXiv2019

On Profitability of Nakamoto double spend

Nakamoto double spend strategy, described in Bitcoin foundational article, leads to total ruin with positive probability and does not make sense from the profitability point of view. The simplest strategy that can be profitable incorporates a stopping threshold when success is unlikely. We solve and compute the exact profitability for this strategy. We compute the minimal amount of the double spend that is profitable. For a given amount of the transaction, we determine the minimal number of confirmations to be requested by the recipient such that this double spend strategy is non-profitable. We find that this number of confirmations is only 1 or 2 for average transactions and a small hashrate of the attacker. This is substantially lower than the original Nakamoto numbers that are widely used and are only based on the success probability instead of the profitability.

cs.CR↗
arXiv2019

Log-euclidean geometry and "Grundlagen der Geometrie"

We define the simplest log-euclidean geometry. This geometry exposes a difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der Geometrie". The list of axioms appears to be incomplete if the foundations of geometry are to be independent of set theory, as Hilbert intended. In that case we need to add a missing axiom. Log-euclidean geometry satisfies all axioms but the missing one, the fifth axiom of congruence and Euclid's axiom of parallels. This gives an elementary proof (with no need of Riemannian geometry) of the independence of these axioms from the others.

math.LO↗
arXiv2019

The Ramificant Determinant

We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus $0$). We define the base vector space of transcendental functions and establish by elementary means some transcendental properties. We introduce the Ramificant Determinant constructed with transcendental periods and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like Theorem and a Torelli-like Theorem. Transposing to the transalgebraic curve the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points.

math.CV↗
arXiv2019

Selfish Mining in Ethereum

We study selfish mining in Ethereum. The problem is combinatorially more complex than in Bitcoin because of major differences in the reward system and a different difficulty adjustment formula. Equivalent strategies in Bitcoin do have different profitabilities in Ethereum. The attacker can either broadcast his fork one block by one, or keep them secret as long as possible and publish them all at once at the end of an attack cycle. The first strategy is damaging for substantial hashrates, and we show that the second strategy is even worse. This confirms what we already proved for Bitcoin: Selfish mining is most of all an attack on the difficulty adjustment formula. We show that the current reward for signaling uncle blocks is a weak incentive for the attacker to signal blocks. We compute the profitabilities of different strategies and find out that for a large parameter space values, strategies that do not signal blocks are the best ones. We compute closed-form formulas for the apparent hashrates for these strategies and compare them. We use a direct combinatorics analysis with Dyck words to find these closed-form formulas.

cs.CR↗