Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates

Conditional prime-pattern results.

Land’s draft gives conditional irrationality [13], and Ringer’s draft gives conditional normality [26]. For each fixed integer base $b\ge 2$, Ringer’s number is $\sum _{n\ge 0}{p}_n{b}^{-(n+1)}$ and its normality is to base $b$. Varying $b$ varies the number, so the result makes no assertion of absolute normality for one constant. The authors supply formalisation material, which we have not rebuilt.

The uniform Hardy–Littlewood hypothesis in [12] counts prime translates of each nonempty admissible tuple of $k\le (\log \log x{)}^3$ distinct integer shifts in $[0,(\log x{)}^2]$. Admissibility means that the shifts omit a residue class modulo each prime. The main term is the singular series of the tuple multiplied by ${\int }_2^x(\log t{)}^{-k}dt$, with absolute error at most $C{x}^{1-\delta }$. The constants $C,\delta >0$ and the starting threshold precede the choice of $x$ and of the tuple. Replacing the integral by its leading asymptotic can exceed this error, and qualitative asymptotics for individual fixed tuples do not give the stated uniformity.

In the version rechecked on 18 September 2026, Ringer also states weaker hypotheses for the joint positions of the first $L$ primes after a prime basepoint, with $L$ of order $\log \log X$. The basepoints are uniform among primes in $(X,2X]$, where $X$ measures prime values. One hypothesis controls a weighted sum of adverse tuple-count errors. The adverse sign alternates with the order of inclusion–exclusion, so both undercounts and overcounts can matter. Another hypothesis allows bounded domination by a comparison law in place of total variation approximation. Neither is a single-gap marginal estimate or a consequence of an upper sieve bound alone. We assume none of these hypotheses here.

Our index band $[X,2X)$ corresponds to $[p_X,p_{2X})$. The prime number theorem gives $\pi (2p_X)=2X+o(X)$, so its symmetric difference with the prime basepoints in $(p_X,2p_X]$ has size $o(X)$. Each set has $X+o(X)$ elements. The two uniform measures have common mass equal to the intersection size divided by the larger cardinality, and hence have total variation distance $o(1)$. This compares the same uniformly bounded observable with the same normalisation at common basepoints. It supplies no relative or quantitative control of the growing patterns required by the preceding conditional results.

Related irrationality problems.

Erdős’s 1958 article states that $\sum _{n\ge 1}{p}_{n-1}^k/n!$ is irrational for each $k\ge 1$, and prints a proof only for $k=1$, explaining that the higher-power proof is more complicated [1]. Schlage-Puchta later proved the stronger linear independence over $\mathbb{Q}$ of $1,S_0,S_1,\dots$, where $S_k=\sum _{n\ge 1}{p}_{n-1}^k/n!$ [18]. The fixed dyadic denominator question is stated separately in [4], including the difficulty of the case $k=1$.

The same page proposes irrationality of $\sum _{n\ge 1}{p}_{n-1}/(d_1\cdots d_n)$ whenever $d_n\ge 2$ and $d_n=o(p_{n-1})$. Kovač’s note of 15 April 2026 disproves this variable-denominator assertion with a sum equal to $1$ [24]. Its printed attribution is ChatGPT 5.4 Pro, orchestrated by Vjeko Kovač. Section 3 of the 1958 article also concerns variable product denominators [1]. We make no quantified reinterpretation of its nonstandard asymptotic notation in condition (5), which is unused here. The endpoint example can be read directly: with $G_0=1$ and $G_n=\prod _{j=1}^n(p_{j-1}+1)$,

These examples leave fixed dyadic denominators unchanged as a separate question.

A different dyadic series with bounded coefficients is due to Erdős and Pomerance. If $P(m)$ is the largest prime factor of $m$, they proved irrationality of

[3, §7, p. 320]. Erdős and Graham record the complementary indicator [2]. Since consecutive integers cannot have the same largest prime factor, its series is $1/2$ minus the displayed one and is also irrational. These binary coefficients differ from the unbounded numerators of $\mathrm{\Pi }$. Schlage-Puchta’s concatenation theorem [18] concerns a base $b\ge 2$ that is not a proper power, with its stated monotonicity and growth assumptions. Concatenation places blocks according to their lengths, whereas our coefficients occupy fixed dyadic positions. We do not use that theorem. The analytic input used below is his Lemma 4, proved by Selberg’s sieve: for every fixed nonzero $F\in \mathbb{Z}[x_0,\dots ,x_k]$, $F(g_n,\dots ,g_{n+k})\ne 0$ for almost all $n$ [18].

Interval covering and divisibility.

The bounded perturbation in Theorem 8.1 is a binary expansion. The sparse construction uses the finite-choice lemma of Crmarić and Kovač [27]. For finite nonnegative choice sets with at least two elements and summable maxima, their lemma fills a single interval if each largest successive gap is at most the sum of the subsequent diameters. Here the sets are $\{0,w_j,\dots ,D_j{w}_j\}$, and the condition is $w_j\le \sum _{i>j}{D}_i{w}_i$. The proof checks this inequality at every stage. Fridy’s lemma [9] assumes nonincreasing weights, a restriction avoided by the finite-choice form. Kovač and Tao use reciprocal choices [11].

Van Doorn and Kovač combine finite subsum covering with a distinguished subsequence of denominators [20]. Each distinguished denominator is divisible by all earlier denominators, and every positive integer divides some denominator. Thus every fixed integer divides all sufficiently late distinguished denominators, as it does our factorial moduli. Their condition is imposed on the distinguished subsequence, not on every denominator. At a distinguished cutoff the finite sums lie in the reciprocal grid of the last denominator. Adding the covering inequalities between successive cutoffs gives Lemma 7 for the full finite prefix, leaving no gaps in that grid. Once the denominator of a rational target divides the last denominator and the target lies in the available range, it has a finite representation. Their proposition fills a half-open interval of rational targets by finite subsums. Our construction fills a fixed interval of real targets by infinite choices.

Van Doorn’s exchange $\{21d,28d\}\leftrightarrow \{20d,30d\}$ preserves the reciprocal sum $1/(12d)$ while changing the ordinary sum from $49d$ to $50d$ [29]. The pair in (1) preserves the ordinary sum and changes the weighted one. The analogy concerns the conserved quantity, and does not invoke his partition theorem. The present argument adds no new interval-covering principle.

Binary subsums.

The achievement set of a positive summable sequence consists of sums obtained by retaining or omitting each term. The sequence $(2^{-n}{)}_{n\ge 1}$ fills $[0,1]$. The sequence $(3^{-n}{)}_{n\ge 1}$ omits $(1/6,1/3)$ because its terms after the first sum to $1/6$. Nevertheless a failure of this simple overlap test at some indices does not exclude interior elsewhere in the achievement set. Bartoszewicz, Filipczak and Szymonik obtain interior from a central run of integer block sums in a multigeometric family [32]. Prus-Wiśniowski and Ptak construct Cantorvals with density-zero overlap indices $\{n:a_n\le \sum _{i>n}{a}_i\}$ [31]. These compact sets are the closures of their interiors, and the endpoints of every nondegenerate interval component are accumulation points of one-point components.

The surrounding classification is discussed in [30] and in Nitecki’s exposition [36], which credits Guthrie and Nymann. The cited preprint’s title and theorem numbers differ from those in the published Monthly article. Even prescribing the exact inequality pattern leaves the presence of interior undetermined. Marchwicki and Miska’s construction [38], with the proof repair in [37], realises any infinite strict term-dominating index set with a Cantor achievement set. The repair preserves uniqueness, and the latter paper gives a simpler proof of a weaker conclusion without uniqueness. Nowakowski’s sufficient condition [39] instead requires every stage of the Star Procedure’s positivity recursion to succeed. Knowing that neither term–tail inequality holds eventually therefore cannot replace the overlap calculation in our variable-digit proof.

Uniformity and finite-state comparisons.

Adamczewski, Drmota and Müllner compute logarithmic densities of each fixed automatic sequence along primes [28]. Their Theorem 1.2 gives a sufficient condition for natural densities to exist and be rational, while Theorem 1.4 reduces logarithmic-density calculation to integers coprime to a fixed modulus. The general logarithmic densities need not be rational. Applying these results to our tails would require a specified finite-alphabet automaton computing the observable, together with uniform error bounds if its state space grows. The theorems themselves concern neither unbounded consecutive gaps nor a growing family of carry automata.

Kuperberg’s singular-series estimate permits a specified growing cardinality [12]. Her arithmetic-progression and smooth-weight estimates [33] provide a fixed-parameter comparison, without giving a law for ordered consecutive-gap blocks. Jha’s revised preprint [34] studies growing short-interval prime counts under a strong Hardy–Littlewood hypothesis. Such counts alone do not control the weighted consecutive-gap differences in our criteria. A fixed-dimensional limit theorem likewise leaves the required growing-block errors to be proved. At the cited revision, the Formal Conjectures repository leaves Problem #251 unproved [25]. Its zero-based sum starts with the zeroth prime divided by $2^0$, and is twice our normalisation.

The tail argument.

An ordinary binary expansion is rational exactly when its digits are eventually periodic. Applying carries to integer coefficients changes the sequence to which this criterion refers. Even bounded nonperiodic coefficients can give a rational sum, as Proposition D.6 shows. A classical alternative is the growth criterion for Ahmes series $\sum _{k}1/a_k$ with strictly increasing positive integer denominators: $a_k^{1/2^k}\to \mathrm{\infty }$ suffices for irrationality [6][11]. Replacing divergence by any sufficiently large fixed threshold fails, since shifted Sylvester sequences grow like $C^{2^k}$ for arbitrarily large $C$ and have rational reciprocal sum. The criterion is sufficient and is not necessary. Splitting $p_i/2^{i+1}$ into $p_i$ equal unit fractions introduces repetitions, and even the distinct denominators $2^n$ satisfy $(2^n{)}^{1/2^n}\to 1$.

We instead track reduced tail denominators. Rationality forces an eventually integral shift, not repeating coefficients: the telescope with $K_n=n$ in Proposition 8.10 has coefficients $n-1$ and sum zero. Parity then gives a sufficient test using two small differences. Its occurrence arbitrarily late for every positive shift remains unproved, as does its necessity.