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Jean-François Burnol

Publications and source records attributed to Jean-François Burnol.

At least 19 recordsLinked to original sources
arXiv2026

Measures for the summation of Irwin series

We consider the series of reciprocals of those positive integers with exactly $k$ occurrences of a given $b$-ary digit $d$ (Irwin series), and obtain geometrically convergent representations for their sums. They are expressed in terms of the moments and Stieltjes transforms of some measures on the unit interval. The moments obey linear recurrences allowing straightforward numerical implementations. This framework also allows to study the large $k$ limit, and, in further works by the author, the large $b$ asymptotics.

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arXiv2026

Full asymptotics and log-periodicity for coefficients in representations of the Riemann zeta function

For each integer $b\ge2$, we obtain representations of the Riemann zeta function in the complex plane as geometrically convergent series. Mellin transform methods and asymptotics for ratios of Gamma functions from the author's recent work allow us to prove, for the coefficients of these series, asymptotic expansions to all orders in inverse powers of the index, with numerators which are periodic functions of the base $b$ logarithm of the index. Such complete asymptotics were first established in the author's recent work for quantities which arose decades ago in theoretical computer science.

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arXiv2026

The modal expansion of Kempner sums

This is a continuation of the author's earlier paper Block-count constrained harmonic sums: spectral expansion and block-directed Euler--Maclaurin. We presented in that reference the modal expansion of block-Irwin sums $I(b,w,k)$ for $k\geq1$. The case $k=0$ was fully developed when $w$ is a single digit, but it was presented only as a very short result summary for $w$ a multi-digit block. Although the framework is already completely exposed in the earlier paper, the case $k=0$ requires adding a separate, dedicated discussion, which is provided here.

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arXiv2026

Block-count constrained harmonic sums: spectral expansion and block-directed Euler-Maclaurin

We determine exactly how harmonic sums, taken over integers with a given number of occurrences of a fixed block of digits, depend on that number of occurrences. Language factorizations, combined with stochastic radix expansions, relate these harmonic sums to the iteration of an operator whose eigenvectors are the Lebesgue measure and distributional derivatives of singular measures. The dual picture is given by a Riesz basis of eigenpolynomials in a suitable Hardy space. The resulting modal expansion admits a natural interpretation as a block-directed Euler-Maclaurin formula interpolating between Euler-Maclaurin and Taylor expansions.

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arXiv2026

On the asymptotics of Kempner-Irwin sums

Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed $d$ and $k$, of $I(b,d,k)-b\log(b)$. We explicitly give, depending on cases, either four or five terms. The coefficients involve the values of the zeta function at the integers.

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arXiv2026

Some series representing the zeta function for $\Re(s)>1$

We represent the Riemann zeta function in the half-plane $\Re s >1$ via series whose terms admit geometrically decreasing bounds. This is based upon simple combinatorics of certain measures, and their associated moments, which are defined on words from an alphabet of digits in a given radix $b$.

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arXiv2026

Some geometric series for Euler's constant

We provide representations of Euler's constant $γ=0.577...$ as series which converge geometrically fast (but use a certain sequence whose computation induces a quadratic cost). The asymptotic oscillations of these coefficients are determined to all orders. A result of independent interest, about sufficient conditions for the validity, in the case of unbounded parameters, for the Tricomi-Erdélyi asymptotic expansion of the ratio of two Gamma functions, is established for that purpose.

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arXiv2026

On the analytic continuation of Dirichlet series with missing digits

We study the Dirichlet series associated with the integers whose radix-$b$ representation misses certain (fixed) digits. The existence of a meromorphic continuation to the entire complex plane, which was already well-known as a general fact valid for $b$-automatic Dirichlet series, is proven anew from a representation as an everywhere defined series with good convergence properties. A generating function related to the residues on the real axis is shown to be the multiplicative inverse of the moment generating function for the associated Cantor set in the unit interval. This makes the (normalized) residues some sort of generalized Bernoulli numbers.

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arXiv2025

Measures associated with certain ellipsephic harmonic series and the Allouche-Hu-Morin limit theorem

We consider the harmonic series $S(k)=\sum^{(k)} m^{-1}$ over the integers having $k$ occurrences of a given block of $b$-ary digits, of length $p$, and relate them to certain measures on the interval $[0,1)$. We show that these measures converge weakly to $b^p$ times the Lebesgue measure, a fact which allows a new proof of the theorem of Allouche, Hu, and Morin which says $\lim S(k)=b^p\log(b)$. A quantitative error estimate will be given. Combinatorial aspects involve generating series which fall under the scope of the Goulden-Jackson cluster generating function formalism and the work of Guibas-Odlyzko on string overlaps.

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arXiv2024

Moments in the exact summation of the curious series of Kempner type

We obtain for the Kempner series (i.e. harmonic series where certain digits are excluded from all denominators, for example the digit 9 in base 10) new representations as geometrically convergent series. The coefficients for these representations involve the power sums on the allowed digits and the moments of an explicitly described measure on the unit interval. These moments can be computed numerically by recurrence. We establish a priori lower and upper bounds for them. This allows converting the theoretical formulas into an efficient numerical algorithm.

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arXiv2024

Digamma function and general Fischer series in the theory of Kempner sums

The harmonic sum of the integers which are missing $p$ given digits in a base $b$ is expressed as $b\log(b)/p$ plus corrections indexed by the excluded digits and expressed as integrals involving the digamma function and a suitable measure. A number of consequences are derived, such as explicit bounds, monotony, series representations and asymptotic expansions involving the zeta values at integers, and suitable moments of the measure. In the classic Kempner case of $b=10$ and $9$ as the only excluded digit, the series representation turns out to be exactly identical with a result obtained by Fischer already in 1993. Extending this work is indeed the goal of the present contribution.

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arXiv2024

Summing the "exactly one 42" and similar subsums of the harmonic series

For $b>1$ and $αβ$ a string of two digits in base $b$, let $K_1$ be the subsum of the harmonic series with only those integers having exactly one occurrence of $αβ$. We obtain a theoretical representation of such $K_1$ series which, say for $b=10$, allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the $K_1$ sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation.

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arXiv2012

Deux extensions de Théorèmes de Hamburger

We propose two types of extensions to Hamburger's theorems on the Dirichlet series with functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary betweeen the moderate meromorphic functions with functional equation and the tempered distributions with extended S-support condition. ----- Nous proposons deux types d'extensions aux théorèmes de Hamburger sur les séries de Dirichlet avec équation fonctionnelle comme celle de la fonction zêta de Riemann, sous des hypothèses plus faibles. Ceci repose sur le dictionnaire entre les fonctions méromorphes modérées avec cette équation fonctionnelle et les distributions tempérées avec la condition de support S-étendue.

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arXiv2011

On the system of the functions zeta(s)/(s-rho)^k

The system of the functions zeta(s)/(s-rho)^k is complete and minimal in a certain sub-Hilbert space of the L^2 space of the critical line. We study whether it is also hereditarily complete.

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arXiv2010

Un déterminant reproduisant

We evaluate the determinant D = det((y_j u_i - x_j v_i)/(l_j - k_i)) as a function of the last sextuple (u,v,k;x,y,l), the result being shown to have a form reproducing the one of the entries of D. ----- Nous calculons le déterminant D = det((y_j u_i - x_j v_i)/(l_j - k_i)) comme fonction du dernier des sextuplets (u,v,k;x,y,l), en exprimant le résultat sous une forme qui reproduit celle des entrées de D.

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arXiv2010

Hilbert spaces of entire functions with trivial zeros

Let $H$ be a Hilbert space of entire functions. Let $H'$ be the space of the functions $f(z)/\prod_i(z-z_i)$ where $f$ belongs to $H$ and vanishes at $n$ given complex points $z_i$. We compute a suitable $E$ function for $H'$ when one is given for $H$.

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