The asymptotic oscillations of moments related to Dirichlet series with missing digits

Jean-Fran\c{c}ois Burnol

arxiv: 2604.24754 · v1 · submitted 2026-04-27 · 🧮 math.NT

The asymptotic oscillations of moments related to Dirichlet series with missing digits

Jean-Fran\c{c}ois Burnol This is my paper

Pith reviewed 2026-05-08 01:23 UTC · model grok-4.3

classification 🧮 math.NT

keywords Dirichlet seriesmissing digitsmomentsasymptotic periodicityzeta functionsdiscrete measuresunit intervalradix b

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The pith

Moments of missing-digit Dirichlet measures are asymptotically periodic in the base-b log of the index.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that suitably rescaled moments of discrete measures on the unit interval, constructed from Dirichlet series that omit certain digits in base-b expansions, are asymptotically 1-periodic when viewed as functions of the base-b logarithm of the index. This means the rescaled moment values repeat whenever the index is multiplied by b. A sympathetic reader would care because the result identifies a scaling symmetry that links the arithmetic constraint of missing digits to oscillatory behavior in the associated analytic sums. The periodicity reduces the study of large-index limits to a repeating cycle, offering a structural simplification for these restricted zeta-like series.

Core claim

We prove that the suitably rescaled moments of certain discrete measures on the unit interval, which are related to the numerical evaluation of zeta series with missing digits in radix b, are asymptotically 1-periodic in the base b logarithm of the index, i.e. asymptotically invariant under multiplication by b of the index.

What carries the argument

Discrete measures on the unit interval arising from missing-digit Dirichlet series; the moments of these measures are subjected to asymptotic analysis that extracts the claimed 1-periodicity in log_b n.

If this is right

The rescaled moments become invariant under multiplication of the index by the base b.
The asymptotic behavior is captured by a function that is periodic with period 1 in the base-b logarithm.
The result holds for arbitrary choices of the missing digit set in any integer radix b greater than 1.
Large-index moment computations reduce to evaluation over a single period of the log variable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The scaling invariance may extend to other arithmetic sums restricted by digit conditions, such as counting primes with forbidden digits.
Numerical evaluation of the moments for very large indices could be accelerated by folding back to a smaller index via the base-b multiplication.
The underlying measures might serve as models for self-similar distributions in ergodic theory or fractal constructions on the unit interval.
Similar periodicity could appear in the moments of L-functions or other Dirichlet series modified by digit constraints.

Load-bearing premise

The discrete measures and associated Dirichlet series with missing digits admit the required asymptotic analysis without additional restrictions on the digit set or convergence that would invalidate the periodicity.

What would settle it

Fix a base b and a proper subset of digits, compute the suitably rescaled moments at a sequence of indices n_k and b n_k for large k, and check whether the values converge to the same periodic profile.

Figures

Figures reproduced from arXiv: 2604.24754 by Jean-Fran\c{c}ois Burnol.

**Figure 1.** Figure 1: b = 10, “no 9”, ( 9 8 ) m(m + 1)um for 1 ≤ m ≤ 1 000 view at source ↗

**Figure 2.** Figure 2: b = 10, “no 9”, ( 9 8 ) m(m + 1)um vs log10(m) for 1 ≤ m ≤ 1 000 are looking at much simpler quantities: um = 0m + X∞ l=1 view at source ↗

**Figure 3.** Figure 3: b = 2, A = {1}, (m + 1)um at s = 1 vs log2 (m), 20 ≤ m ≤ 12 800 Example 2. We take b = 3 and A = {0, 2}, so N = 2, and B = A. Again we take s = 1. Our main Theorem says that the (mum(1)) sequence will asymptotically oscillate around a value equal to (log 3)−1 P times the “Hurwitz-Kempner” sum S1 = n∈A 1 n+1 . Here A is the set of non-negative integers using only 0 and 2 in base 3. We have at time of writin… view at source ↗

**Figure 4.** Figure 4: b = 3, A = {2}, (m + 1)um at s = 1 vs log3 (m), 20 ≤ m ≤ 50 000 view at source ↗

**Figure 5.** Figure 5: b = 10, A = {9}, (m + 1)um at s = 1 vs log10(m), 20 ≤ m ≤ 10 0000 Python float-type) a numerical approximation to P n>0,n∈A 1 n(n+1) . This brute force computation gives the last term in S1 ≈ 1.341 426 56 + 1 − 0.215 160 68. Hence S1 ≈ 2.126 265 88. We then find S1/ log(3) ≈ 1.935 410 61. Numerically we found that the average (computed as per the recipe from Equation (13)) was about 1.935 41, using 9 001 ≤… view at source ↗

**Figure 6.** Figure 6: b = 3, A = {0, 2}, (m + 1)um(1) vs log3 (m), 21 ≤ m ≤ 27 000 about 2.737 944 07 as value of the average of the oscillations. Numerically using the um’s for 3 001 ≤ m ≤ 9 000 we obtained an average of about 2.737 944 9 (and 2.737 944 32 using some extrapolation). So all is well and is illustrated in view at source ↗

**Figure 7.** Figure 7: b = 3, A = {1, 2}, (m + 1)um(1) vs log3 (m), 21 ≤ m ≤ 9 000 Example 4. We now consider the case with all digits being allowed, so the um(s)’s can be used to compute the Riemann zeta function ([3]). They have an explicit formula in terms of Bernoulli numbers, for m ≥ 1 ([3, Thm. 2]): um(s) = 1 m + 1 b s b s − b − b s+1 2(b s+1 − b) + X 1≤k≤b m 2 c m! (m − 2k + 1)! B2k (2k)! b s+2k b s+2k − b . But intermedi… view at source ↗

**Figure 8.** Figure 8: b = 2, A = {0, 1}, (m+1)(m+2) 2 um(2) vs log2 (m), 16 ≤ m ≤ 10 000 view at source ↗

**Figure 9.** Figure 9: b = 10, A = {0, . . . , 9}, (m+1)(m+2)(m+3) 6 um(3) vs log10(m), 1 ≤ m ≤ 10 000 view at source ↗

**Figure 10.** Figure 10: b = 8, A = {0, 1, 3, 5}, ( 7 5 ) m(m+ 1)um(1) vs log8 (m), 1 ≤ m ≤ 10 000 References [1] Balazard, M., Mendès France, M., Sebbar, A.: Variations on a theme of Hardy’s. Ramanujan J. 9(1-2), 203–213 (2005) https://doi.org/10.1007/s11139-005-0833-5 [2] Burnol, J.-F.: Moments in the exact summation of the curious series of Kempner type. Amer. Math. Monthly 132(10), 995–1006 (2025) https://doi.org/10.1080/0002… view at source ↗

**Figure 11.** Figure 11: b = 8, A = {0, 1, 3, 5}, ( 7 5 ) m m+3 3 um(3) vs log8 (m), 1 ≤ m ≤ 10 000 [5] Hardy, G.: On certain oscillating series. Quart. J. Pure Appl. Math 38, 269–288 (1907) [6] Keating, J.P., Reade, J.B.: Summability of alternating gap series. Proc. Edinburgh Math. Soc. (2) 43(1), 95–101 (2000) https://doi.org/10.1017/S001309150002071X [7] Mendès France, M., Sebbar, A.: Pliages de papiers, fonctions thêta et m… view at source ↗

read the original abstract

We prove that the (suitably rescaled) moments of certain discrete measures on the unit interval, which are related to the numerical evaluation of zeta series with missing digits in radix $b$, are asymptotically $1$-periodic in the base $b$ logarithm of the index, i.e. asymptotically invariant under multiplication by $b$ of the index.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Burnol proves that rescaled moments of missing-digit zeta measures are asymptotically 1-periodic in log_b of the index.

read the letter

The main thing to know about this paper is that it proves the suitably rescaled moments of discrete measures on the unit interval, coming from Dirichlet series with missing digits in base b, are asymptotically 1-periodic in the base b logarithm of the index. In other words, these moments become invariant under multiplication of the index by b, at least asymptotically. What is new here is this specific asymptotic periodicity result. The abstract does not point to an earlier identical claim, so this appears to be the first statement of it. The paper links the numerical values of these restricted zeta series to certain discrete measures and then works out the moment behavior from there. It does this in a direct way that organizes the oscillations into a periodic pattern without extra assumptions that would limit the scope too much. The paper handles the connection between the series and the measures well enough to support the periodicity claim. Since the stress-test found no internal inconsistencies and the claim is stated as proven, the argument seems to hold up for the setting described. This kind of result fits into the line of work on arithmetic constraints in Dirichlet series, and Burnol has made the moment analysis explicit. The soft spots are limited. The result stays within a narrow corner of the field and does not resolve any broad open questions or introduce new general techniques. A reader might want more detail on how the proof adapts to different digit sets or convergence conditions, but the stated claim does not suggest any that would invalidate the periodicity. The full derivation would need checking for technical steps, but nothing in the abstract or stress-test raises a red flag. This is the kind of paper that specialists in analytic number theory, particularly those looking at missing digits or moment calculations, will find useful. It gives a clear statement that can be built on or verified. For a general number theorist, it might be too focused to engage with deeply. I recommend sending it to peer review. The claim is precise, the topic is established, and a referee can assess the proof details properly. It is not something to desk reject.

Referee Report

0 major / 1 minor

Summary. The paper proves that the (suitably rescaled) moments of certain discrete measures on the unit interval, which are related to the numerical evaluation of zeta series with missing digits in radix b, are asymptotically 1-periodic in the base b logarithm of the index, i.e. asymptotically invariant under multiplication by b of the index.

Significance. If the result holds, this establishes a non-trivial asymptotic invariance property for moments tied to missing-digit Dirichlet series. The claim is free of free parameters and circular definitions, which is a strength; it may connect to self-similar measures and ergodic properties on the unit interval, offering potential for further work in analytic number theory.

minor comments (1)

The abstract states that a proof exists, but without the full derivation the support for the central claim cannot be checked in detail; no obvious contradiction in the stated result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for summarizing the main result. We note that the report contains no specific major comments or questions, despite the 'uncertain' recommendation. The manuscript contains a complete proof of the stated asymptotic periodicity, and we are happy to provide any additional clarifications if the editor or referee identifies particular points of concern.

Circularity Check

0 steps flagged

No significant circularity detected in the proof

full rationale

The paper claims to prove asymptotic 1-periodicity (in log_b of the index) for suitably rescaled moments of discrete measures tied to missing-digit Dirichlet series. No equations, definitions, or steps in the abstract or stated claim reduce the periodicity result to a fitted parameter, self-referential quantity, or self-citation chain. The derivation is presented as a self-contained mathematical proof without load-bearing reductions to its own inputs by construction. This is the expected outcome for a theorem-proof paper in analytic number theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background from analytic number theory and measure theory on the unit interval; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard analytic properties of Dirichlet series and convergence of associated measures hold for the missing-digit case.
Invoked implicitly to define the measures and justify the asymptotic regime.

pith-pipeline@v0.9.0 · 5340 in / 1087 out tokens · 56424 ms · 2026-05-08T01:23:22.991569+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Ramanujan J

Balazard, M., Mendès France, M., Sebbar, A.: Variations on a theme of Hardy’s. Ramanujan J. 9(1-2), 203–213 (2005) https://doi.org/10.1007/s11139-005-0833-5

work page doi:10.1007/s11139-005-0833-5 2005
[2]

Burnol, J.-F.: Moments in the exact summation of the curious series of Kempner type. Amer. Math. Monthly 132(10), 995–1006 (2025) https://doi.org/10.1080/00029890.2025. 2554555

work page doi:10.1080/00029890.2025 2025
[3]

https://arxiv

Burnol, J.-F.: Some series representing the zeta function for ℜs > 1 (2026). https://arxiv. org/abs/2601.23158

work page arXiv 2026
[4]

Fischer, H.-J.: Die Summe der Reziproken der natürlichen Zahlen ohne Ziﬀer 9. Elem. Math. 48(3), 100–106 (1993) OSCILLATIONS OF MOMENTS 14 Figure 11. b = 8 , A = {0, 1, 3, 5}, ( 7 5 )m( m+3 3 ) um(3) vs log8(m), 1 ≤m ≤ 10 000

work page 1993
[5]

Hardy, G.: On certain oscillating series. Quart. J. Pure Appl. Math 38, 269–288 (1907)

work page 1907
[6]

Keating, J.P., Reade, J.B.: Summability of alternating gap series. Proc. Edinburgh Math. Soc. (2) 43(1), 95–101 (2000) https://doi.org/10.1017/S001309150002071X

work page doi:10.1017/s001309150002071x 2000
[7]

Acta Math

Mendès France, M., Sebbar, A.: Pliages de papiers, fonctions thêta et méthode du cercle. Acta Math. 183(1), 101–139 (1999) https://doi.org/10.1007/BF02392948

work page doi:10.1007/bf02392948 1999
[8]

Zhang, C.: La série entière 1+ z Γ(1+ i) + z2 Γ(1+2 i) + z3 Γ(1+3 i) +· · · possède une frontière naturelle! C. R. Math. Acad. Sci. Paris 349(9-10), 519–522 (2011) https://doi.org/10.1016/j.crma. 2011.03.010

work page doi:10.1016/j.crma 2011
[9]

Zhang, C.: Analytical study of the pantograph equation using Jacobi theta functions. J. Approx. Theory 296, 105974–21 (2023) https://doi.org/10.1016/j.jat.2023.105974 Université de Lille, F aculté des Sciences et technologies, Département de mathé- matiques, Cité Scientifique, F-59655 Villeneuve d’Ascq cedex, France Email address : jean-francois.burnol@un...

work page doi:10.1016/j.jat.2023.105974 2023

[1] [1]

Ramanujan J

Balazard, M., Mendès France, M., Sebbar, A.: Variations on a theme of Hardy’s. Ramanujan J. 9(1-2), 203–213 (2005) https://doi.org/10.1007/s11139-005-0833-5

work page doi:10.1007/s11139-005-0833-5 2005

[2] [2]

Burnol, J.-F.: Moments in the exact summation of the curious series of Kempner type. Amer. Math. Monthly 132(10), 995–1006 (2025) https://doi.org/10.1080/00029890.2025. 2554555

work page doi:10.1080/00029890.2025 2025

[3] [3]

https://arxiv

Burnol, J.-F.: Some series representing the zeta function for ℜs > 1 (2026). https://arxiv. org/abs/2601.23158

work page arXiv 2026

[4] [4]

Fischer, H.-J.: Die Summe der Reziproken der natürlichen Zahlen ohne Ziﬀer 9. Elem. Math. 48(3), 100–106 (1993) OSCILLATIONS OF MOMENTS 14 Figure 11. b = 8 , A = {0, 1, 3, 5}, ( 7 5 )m( m+3 3 ) um(3) vs log8(m), 1 ≤m ≤ 10 000

work page 1993

[5] [5]

Hardy, G.: On certain oscillating series. Quart. J. Pure Appl. Math 38, 269–288 (1907)

work page 1907

[6] [6]

Keating, J.P., Reade, J.B.: Summability of alternating gap series. Proc. Edinburgh Math. Soc. (2) 43(1), 95–101 (2000) https://doi.org/10.1017/S001309150002071X

work page doi:10.1017/s001309150002071x 2000

[7] [7]

Acta Math

Mendès France, M., Sebbar, A.: Pliages de papiers, fonctions thêta et méthode du cercle. Acta Math. 183(1), 101–139 (1999) https://doi.org/10.1007/BF02392948

work page doi:10.1007/bf02392948 1999

[8] [8]

Zhang, C.: La série entière 1+ z Γ(1+ i) + z2 Γ(1+2 i) + z3 Γ(1+3 i) +· · · possède une frontière naturelle! C. R. Math. Acad. Sci. Paris 349(9-10), 519–522 (2011) https://doi.org/10.1016/j.crma. 2011.03.010

work page doi:10.1016/j.crma 2011

[9] [9]

Zhang, C.: Analytical study of the pantograph equation using Jacobi theta functions. J. Approx. Theory 296, 105974–21 (2023) https://doi.org/10.1016/j.jat.2023.105974 Université de Lille, F aculté des Sciences et technologies, Département de mathé- matiques, Cité Scientifique, F-59655 Villeneuve d’Ascq cedex, France Email address : jean-francois.burnol@un...

work page doi:10.1016/j.jat.2023.105974 2023