Papers

The secondary zeta function is defined by $$ Z(s)=\sum_{\gamma>0}\frac{1}{\gamma^{s}}, $$ where $\tfrac12+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of $Z(s)$ at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann $\xi$-function. Define $$ L(x)=\log\left(\frac{\xi(x+\tfrac12)}{\xi(\tfrac12)}\right), \qquad \mathcal{E}=x\frac{\mathrm{d}}{\mathrm{d}x}, \qquad \mathcal{D}_k=\mathcal{E}\prod_{j=1}^{k}(\mathcal{E}-2j). $$ Then, for $k\geq1$, $$ Z(2k)=\frac{(-1)^k}{2^k(k-1)!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+1}}\,\mathrm{d}x $$ and $$ Z(2k+1)=\frac{(-1)^k}{\pi\,(2k-1)!!}\int_{0}^{\infty}\frac{\mathcal{D}_k L(x)}{x^{2k+2}}\,\mathrm{d}x. $$ In the last part, we write these formulas out explicitly for $Z(k)$, $2\leq k\leq 7$, both in terms of $L(x)$ and in terms of $\xi(x+\tfrac12)$ for reference. We also derive a limit formula for $\gamma_1$, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.

Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.

We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving $25$ out of $30$ two-variable equations of size up to $34$ for which the problem of integer solubility remained open. We also resolve the remaining $5$ equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to $35$ and of length up to $12$, with two exceptions. Before this work, there were $211$ equations in this category that resisted the previously known methods, and we proved that $209$ of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.

A continuous shape parameter enters both a radial density and the normalization of its transform coefficients. This coupling allows changes in shape to be compensated by changes in scale and exponential tilt. We study the coefficients b_k=e^a E[Y^k ]/[k!(d)_k ] for densities proportional to y^(d-1) exp(-{\alpha}y^q+e̱ta}y), with known integer q>=2, d,{\alpha}>0, and unrestricted real a,e̱ta}. For q=2, the first four coefficients determine all parameters globally, whereas three coefficients leave an analytic curve of solutions. We characterize the exact image of the two shape ratios and prove that two alternating corners decide whether an entire coefficient box is realizable and give its exact shape range. For every fixed q, the first q(q+1) coefficients give global identification and an everywhere full-rank map. A finite-difference criterion yields generic identification from 2q+1 coefficients, including every zero-tilt model against the full tilted family. For q=3, an exact rational certificate constructs two distinct shapes with identical first four coefficients and separated fifth coefficients. The results distinguish exact identification, realizability, and sensitivity for a parameter-dependent coefficient normalization.

A set of $m$ distinct positive integers is called a Diophantine $m$-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple $\{a,b,c,d\}$ with $a<b<c<d$ is called regular if $$d=a+b+c+2abc+2\sqrt{(ab+1)(ac+1)(bc+1)}.$$ We prove that every irregular Diophantine quadruple satisfies $c<10^{338}$ and $d<10^{10^{24}}$.

The 2026 measles resurgence in Bangladesh motivates a spatial modelling framework that accounts for heterogeneous local transmission, vaccination activity, and interdivisional connectivity. We formulate a multipatch susceptible-vaccinated-infectious-recovered model for the eight administrative divisions of Bangladesh. Division-specific parameters are estimated from reported measles cases \cite{DGHS} and smoothed vaccination data using bounded nonlinear least squares, with uncertainty quantified through 1000 parametric bootstrap replicates. The fitted divisional systems are then connected through compartment-specific movement and cross-patch infection coefficients adjusted for connectivity and the source division's relative reproduction number. The fitted model captures the main division-specific epidemic patterns. Estimated local reproduction numbers range from \(1.1386\) in Khulna (95% CI:\(0.5703\)–\(1.3834\)) to \(4.3491\) in Dhaka (95% CI: \(1.4449\)–\(5.6307\)), while Chattogram also exhibits high transmission potential (\(\mathcal{R}_0^i=4.2275\); 95% CI: \(4.0019\)–\(4.3185\)). Baseline spatial coupling increases the network-wide epidemic peak from \(190.96\) to \(313.20\) infectious individuals and more than doubles cumulative infections from \(1086.42\) to \(2345.35\). Relative to baseline coupling, 50% and 80% restrictions reduce the epidemic peak by \(26.69\%\) and \(34.37\%\), and cumulative infections by \(25.07\%\) and \(41.48\%\), respectively. Conversely, a 50% increase in spatial coupling raises the peak by \(30.14\%\) and cumulative infections by \(23.54\%\). Although local transmission remains dominant in every division, cross-patch infection accounts for \(28.83\%\) of new infections in Rajshahi at the baseline epidemic peak. Source-sink analysis identifies Dhaka as the dominant cross-patch source and Chattogram as a smaller secondary source.

Let $K$ be an imaginary quadratic field with fundamental discriminant $D=-5n$, where $n>0$ is odd, and cyclic ideal class group $\text{Cl}(K)=\langle g\rangle\simeq C_{4M}$. Assume that the ramified prime class above $5$ is the unique involution $g^{2M}$. For the Kronecker eta invariant \[ F(A)=\frac{|\eta(z_A)|^2}{\sqrt{a_A}}, \] where $z_A$ and $a_A$ arise from a positive definite quadratic form representing the ideal class $A$, we show that the logarithms of the mirror ratios \[ R_j=\frac{F(g^{2M-j})}{F(g^j)},\qquad 0\le j<M, \] are related by an explicit discrete cosine transform to the odd class-character Hecke $L(1)$-values. A single level-$5$ generalized-Weber orbit realizes all of these mirrors. Combining Fourier nonvanishing, lower-power Weber class invariants and a ray-class cancellation argument, we prove \[ \mathbb{Q}(R_j)=H^{\langle c\rangle},\qquad R_j^{\sigma^{2M}}=R_j^{-1},\qquad \mathbb{Q}(R_j+R_j^{-1})=H^{\langle c,\sigma^{2M}\rangle}. \] Here $H$ is the Hilbert class field, $c$ denotes complex conjugation, and $\sigma$ corresponds to $g$ under the Artin map. Consequently \[ [\mathbb{Q}(R_j):\mathbb{Q}]=4M,\qquad [\mathbb{Q}(R_j+R_j^{-1}):\mathbb{Q}]=2M, \] and the latter degree becomes $M$ over the real genus field $\mathbb{Q}(\sqrt5)$. Thus the degree pattern $2^r,2^{r-1},2^{r-2}$ in the cyclic $2$-power hierarchy is a specialization of a uniform $C_{4M}$ theorem. We give exact low-class-number evaluations and certified computations at $D=-895,-3295,-8495$, separating respectively four, eight and sixteen non-genus Hecke $L(1)$-values and producing corresponding alternating Rogers-Ramanujan singular values. Classical mixed modular equations then give ordinary Rogers–Ramanujan companions at the same arguments.

For an odd prime p, we study the change in matrix order under lifting from Z/p^k Z to Z/p^(k+1) Z. The first variation of the order defect is an affine translate of a cyclic adjoint transfer on sl_2(F_p). We classify the full-order transfer for every q in GL_2(F_p): its rank is 3 for scalar elements, 1 for nonscalar semisimple elements, 1 for nonscalar Jordan elements when p = 3, and 0 for nonscalar Jordan elements when p > 3. For a Jordan element q = lambda(I + U), U^2 = 0 and U != 0, the vanishing first-order transfer for p > 3 admits an exact higher-order refinement. Writing d = (p - 1)/2, the full-order perturbation has depth d when tr(UX) != 0, depth p - 1 when tr(UX) = 0 and X is not in the span of U, and vanishes identically for X in the span of U. For p = 3, among the 27 lifts of a nonscalar order-preserving parent with determinant fixed to +1 or -1, exactly 9 preserve its order and 18 triple it; after the first increase, every further lift triples the order. We also give a scalar obstruction criterion for order preservation among unrestricted Jordan lifts.

We give a self-contained derivation of the closed form of the power sums $S_m(n)=\sum_{i=1}^{n}(a_0+(i-1)d)^m$ in which the Bernoulli numbers enter as constants of integration rather than as prescribed coefficients. The derivation rests on an elementary fact: a sum whose terms are polynomial in the index is the restriction to the integers of a unique polynomial, which satisfies the defining difference equation identically and may therefore be differentiated and integrated exactly. Differentiating lowers the power by one; integrating back introduces a single constant, equal to the slope $S_m'(0)$ of the polynomial at the origin and fixed by the value $S_m(1)=a_0^m$. This yields a closed form for $S_m(n)$ in terms of the constants and a recurrence that determines them, obtained without induction on the exponent and without generating functions. With the Bernoulli numbers defined by their classical recurrence, we identify the constants as $d^mB_m(a_0/d)$; the progression starting at $0$ reproduces that recurrence exactly. In the normalized case they are $B_m(1)$, so $C_1=B_1(1)=+\tfrac12$, a value that reflects the summation range; a reflection identity shows, without generating functions, that the odd constants beyond the first vanish. The coefficients of the recurrence are scaled binomial coefficients from a single row of Pascal's triangle, and the same approach gives closed forms for sequences with polynomial differences. The integral recursion behind the method is known in the normalized case; each result is related to its classical counterpart, and a verified implementation is provided.

The Diophantine equation $F_n^x+F_m^y=2^a,$ where $F_k$ denotes the $k$-th Fibonacci number and $n,m,a\in \mathbb{Z}_{\geq0}$, has attracted attention in recent years. The cases $(x,y)=(1,1)$ and $(x,y)=(2,2)$ were completely resolved by Bravo and Luca, and by Jeong and Park, respectively. In this note, we investigate the mixed exponent case $(x,y)=(2,1)$ and we completely determine all nonnegative integer solutions.

Ordinary angles and unsigned areas discard orientation, and much of the case analysis in plane geometry is the price of that loss. Signed quantities (directed angles, oriented areas and directed distances) keep the signs and so let one argument serve for every configuration. We illustrate this with results that hold uniformly for convex, concave and crossed quadrilaterals: the diagonal area formula, an identity expressing one diagonal in terms of the sides and the other diagonal, a test for a pair of parallel sides, a theorem on equidiagonal quadrilaterals (such a quadrilateral has a pair of equal opposite sides if and only if it has a pair of parallel opposite sides), and a single ratio of directed distances that decides whether the vertices lie on an ellipse, only on hyperbolas, or on a circle.

We study a family of special numbers and polynomials introduced by Simsek whose exponential generating function has as denominator a falling factorial in the exponential variable. Using the partial fraction decomposition of this denominator, each member of the family is expressed by a single finite sum involving Bernoulli polynomials, Apostol Bernoulli polynomials and two variable Fubini polynomials evaluated at reciprocals of integers. A second expansion, around the point where the denominator vanishes, expresses the same numbers through Stirling numbers of the second kind and generalized harmonic numbers, and yields closed formulas for the first few members. We construct a function, built from the Hurwitz zeta function and Lerch s transcendent, which interpolates the polynomials at the negative integers and gives a closed form of a multiple series recently introduced by Pokaz, Ribicic Penava and Simsek, we give an Euler type integral representation for it, and show that the numbers are signed combinations of moments of geometric random variables. Recurrence relations, asymptotics and an extension to arbitrary distinct nodes are also given.

We construct nonnegative bounded compactly supported functions on the real line that give the lower bound $\mathcal{A}\ge 0.4103779774321214877852987\ldots$ for the minimum-autocorrelation constant. This improves the explicitly certified bound $2378625/5958277=0.39921356\ldots$ reported by Russell. The construction combines a point mass with a monotone degree-ten polynomial density; a one-sided atom-removal argument then produces ordinary functions. Its numerical content reduces to exact rational polynomial identities and 20 Bernstein subinterval certificates. The finite certificate, actual overlap integrals, admissibility, mass formula, and ordinary-function witness bounds have been formalized in Lean 4.19.0; the pinned proof sources and recorded audit are supplied. The passage to the supremum constant is given separately as a mathematical argument. As applications, the same construction yields an upper bound below $1.5610183$ for the large-multiplicity generalized-difference-basis coefficient and a feasible squared mass below $2.436778$ for a continuous sparse-ruler relaxation. An appendix quantifies the loss under independent sampling followed by completion. No improved bound for the complete sparse ruler constant, or global optimality of the construction, is asserted.

We investigate the zeros of Leonardo polynomials and their distribution in the complex plane. We show that the structure of the zeros differs between even and odd degrees. For even degrees, all zeros are simple and purely imaginary, with imaginary parts lying in \((-2,2)\). For odd degrees, there is exactly one positive real zero, whereas no negative real or purely imaginary zeros occur. We obtain two-sided bounds for the unique positive real zero, prove that it decreases strictly to zero as the degree increases through odd values, and determine its asymptotic behavior. Using the relation between Leonardo and Fibonacci polynomials together with a complex trigonometric parametrization, we transform the zero equation into a family of fixed-point problems. Suitable invariant disks are constructed, and the corresponding mappings are shown to be contractions. This yields a constructive description of the complete zero set, proves the simplicity of all zeros, and provides geometrically convergent iterations for their numerical approximation. The method also provides explicit control over the approximation error. Finally, we establish a degree-dependent vertical strip containing all zeros and show that its width tends to zero as the degree increases. Consequently, the zeros become uniformly concentrated near the imaginary axis. Numerical examples illustrate the convergence of the proposed method, the associated error estimates, and the resulting zero distributions.

We study the length of one round of Uno under the official rules, for N players who each start with M cards. The fastest possible round takes at least M turns with two players and at least 2M-1 turns with three or more, and both bounds are attained for small M. An exact search over random deals shows that real deals rarely allow them. Under the official rules a round has no maximum length. Under forced play the probability that a round lasts more than t turns decays exponentially in t at a rate that does not depend on M. Our main result concerns very large hands. If every card a player ever holds is eventually played, a round lasts (27/19)NM turns to first order, and the constant depends only on the composition of the deck. The official rule that a Wild Draw Four may only be played by a player who holds no card of the current colour breaks this law. Every large hand hoards Wild Draw Fours, M/23 of them per player, and the round ends with a dumping endgame. For an even number of players this gives (53/46)NM turns. For an odd number of players the dumping cannot alternate round the table, and the endgame settles into a seating pattern whose turn rate we compute numerically. Simulations with up to 32,000 cards per hand support these limits. Finally we solve a small strategy game: with two players the Nash equilibrium over a family of 48 strategies is a single pure strategy that wins 62.8% of rounds against uniformly random play.

In this article, we mainly investigate when the binary operation $T^{[n]}:[0, 1]^2\rightarrow [0, 1]$ satisfies the associative law and thus becomes a triangular norm where \begin{equation*} T^{[n]}(x,y)=\left\{\begin{array}{ll} f_{n}(T(f_{n}(x),f_{n}(y))), &\text{if}\,\, (x,y)\in{[0,1)^{2}}, \\ \min\{x,y\}, &\text{otherwise,} \end{array}\right. \end{equation*} $f_{n}:[0,1]\rightarrow\left\{0, \frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1\right\}$ given by $f_{n}(x)=\frac{1}{n}{\lfloor nx \rfloor}$ with $n$ a positive integer and $T:[0, 1]^2\rightarrow [0, 1]$ is a triangular norm. We present some sufficient conditions for $T^{[n]}$ to be triangular norms, and for $T^{[n]}$ to be never associative when $T$ is strictly monotonic.

Paulsen and West associate perfect cuboids with triples of rational points on a congruent-number elliptic curve and assert that the corresponding curve has rank at least two. The proof of their Theorem 4.2 contains a gap: translation of a generator by a point of order two interchanges only the odd-indexed terms of two coordinate sequences, rather than the entire sequences. We exhibit two explicit counterexamples to that intermediate assertion and give a complete proof of the rank obstruction. The division-polynomial argument reduces a hypothetical rank-one configuration to one remaining parity pattern. Ingram's primitive-divisor theorem then forces an additive relation among its indices, which is incompatible with a positivity lemma on the elliptic curve. More precisely, the three points of every positive cuboid configuration generate a subgroup of rank at least two, even when the ambient curve has larger rank. The proof is unconditional.

We give a complete local proof of Langlands's First Main Lemma for local epsilon factors over nonarchimedean local fields. The proof builds on Dwork's $p$-adic calculations and Langlands's formulation of the local method. The present paper reformulates these calculations in a common normalization and extends the argument to equal characteristic. A more systematic treatment of stationary parameters and conductor cases is intended to make the calculations easier to follow. Norm–trace correction formulas yield the equal-characteristic extension. The author is also completing a local proof of Langlands's Second Main Lemma and will verify it in Lean. Together with the present result and Langlands's remaining reductions, that work will give a complete proof of the local construction of local epsilon factors envisioned by Langlands. OpenAI ChatGPT assisted with drafting and wording revisions of this manuscript; the author audited the manuscript, checked the proof, and assumes full responsibility.

In this paper, by using a signs rule for a class of special power series and a recurrence relation of coefficients involving hypergeometric series, we find the necessary and sufficient conditions for the sharp (at the left endpoint) inequalities between the hypergeometric function and power mean (Hm-Pm inequality) to hold in all possible cases, which greatly extends those known results and solve the rest of problem on the sharp (at the left endpoint) power mean bounds for the hypergeometric mean.

Fractional integral operators involving special functions as kernels have long been one of the main subjects of fractional calculus. As a discrete version of classical fractional calculus, $q$-fractional calculus not only brings often new phenomena and new perspectives but also provides new useful areas for describing and studying new problems of applied nature. Recently, Luo and Zhou (see below) developed a methodology for formulating well-behaved $q$-fractional integral operators steming from a basic general theory. Our aim in this paper is to define and introduce a new class of $q$-fractional integral operators whose kernel involves a new multivariate $q$-Mittag-Leffler function. We shall also define the corresponding $q$-fractional derivatives (as the inverses of $q$-fractional integral operators) and define also the $q$-Caputo fractional derivative operators. We discuss in detail the similarities and variations between the $q$-operators and their counterparts in classical theory, particularly in Prabakar fractional calculus. Further, applications of the defined operators in solving certain $q$-fractional differential equations are also considered. We also mention throughout this paper some useful relevances of our established results with certain known important cases. In the concluding remarks, we point out briefly few possible proposals (or directions) for further work on the present subject matter of $q$-calculus.

Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets $S\subseteq\N$, the series $-\sum\mu_{\mathcal P}(\lambda)q^{|\lambda|}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\to1$. We determine when this happens. Radially, the summation kernel is a Gumbel density, and Wiener's theorem shows that for bounded $f$ the associated series tends to $L$ if and only if $f$ has mean $L$ on every interval $(x,x+cx/\log x]$; natural density does not suffice. Multiplied by $\log(1/(1-q))$, the series tends to $\delta$ if and only if $S$ has $(c\delta+o(1))x/(\log x)^2$ elements in each such interval; this gives a partition analogue of Alladi's theorem for Chebotarev sets of primes. Off the real axis the largest weights grow like $e^{\Psi_\beta/t}$ with $\Psi_\beta>0$. Along $q=e^{-t(1+i\beta)}$, the formula for residue classes modulo $m\ge3$ holds exactly when $|\beta|<\beta_m$, an explicit Clausen-function threshold with $\beta_m\to0$, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.

We show that for certain classes of matrices over octonions, a weak version of breakthrough numerical radius inequalities obtained by Kittaneh [\textit{{Stud. Math.}, 2005}] hold.

In this article we examine a problem stated in Grübaum and Shephard's classic book on tiling theory, Tilings and Patterns, which asks the reader to find all unilateral and equitransitive tilings by squares of sizes $1, 2, 3, \ldots, n$ where $n \in \mathbb{N}$. The unilaterality condition requires that two squares of the same size in such a tiling may not intersect along an entire side, and the equitransitivity requires that two tiles of the same size in such a tiling must be in the same transitivity class with respect to the symmetry group of the tiling. These two conditions on tilings by squares allows for surprisingly interesting and complex tilings by squares. In this work we provide the answer to Grübaum and Shephard's question for the case $n = 4$, as well as developing methods and computer algorithms that may be extended to the cases where $n \geq 5$.

For decades, university mathematics assessed executable procedure, computing and reproducing standard proofs, as a proxy for the understanding it cared about, because procedure was cheap to mark and understanding was not. Generative AI now executes much of this procedure. It does not merely let students bypass the proxy; it exposes how much of what was certified was a proxy. The crisis is not simply cheating but a proxy problem, and resolving it forces a question the discipline has long avoided: what is the irreducible human core of mathematical competence, and how can it be built when some of the executable work can be offloaded? We set out two futures: displacement-and-re-centering, in which AI takes over some execution and education moves toward judgment, formulation and verification; and atrophy, in which premature offloading may erode capacities that depend on prior construction. At the heart of the optimistic future is a verification instability: some forms of oversight are cheap and bounded, while others approach the competence required to construct the work. The outcome is a design choice, not a technological destiny. Rather than add a model, we state the response as two principles: construct before you delegate, and let verification set the ceiling: how far a student can independently verify AI output limits what they may safely delegate. We illustrate with a sequence and two worked problems, and close on the question: whether the construct-before-delegate dependency is a hard constraint or an artefact of how mathematics has been taught.

For a bounded operator $T$ between Banach spaces, let $d_+(T)$ be its distance to the operators that are not upper semi-Fredholm, and let $\inn(T)$ be Schechter's quantity $\inf\|T|_M\|$ over infinite-dimensional subspaces $M$. One always has $\inn(T)\le d_+(T)$, and González and Martinón showed that the two quantities are not equivalent in general. We prove that $d_+(T)$ is the infimum of the norms of operators on the whole space that extend a restriction of $T$ to some infinite-dimensional subspace. The gap between $\inn$ and $d_+$ thus becomes an extension problem, and projection constants enter naturally. If the domain is uniformly subprojective with constant $\lambda$, then $d_+\le\lambda\,\inn$. In the other direction we prove a quantitative obstruction: a complemented Hilbertian subspace all of whose infinite-dimensional subspaces have projection constant at least $\Lambda$ produces operators with $d_+/\inn\ge\Lambda/2$. Consequently, on the subprojective but not uniformly subprojective space $(\bigoplus_nL_{p_n})_{\ell_2}$ with $p_n\to\infty$, there is no uniform comparison between $\inn$ and $d_+$. We conjecture that uniform comparison is equivalent to uniform subprojectivity. Finally, we compute $\inn$ for inverses and for shears in terms of the quantity $\tau(T)=\sup_M j(T|_M)$, and we use the arbitrary distortability of Hilbert space to show that a rectangular diagonal operator $\operatorname{diag}(A,B)$ built from two isomorphisms with $\inn(A)=\inn(B)=1$ can have arbitrarily small $\inn$. In contrast, every operator $T$ on a complex Banach space satisfies $\inn(T)\le\tau(T)$, so every automorphism satisfies $\inn(T)\,\inn(T^{-1})\le1$ and the same mechanism cannot work for endomorphisms.

We study a divisor-weighted version of Golomb's self-description: the nondecreasing sequence $A$ in which each positive integer $m$ occurs $\sum_{d\mid m}A(d)$ times. This rule has a unique solution, and $$ A(n)\sim Cn^{\varphi-1},\quad C=\left(\frac{\varphi}{\zeta(\varphi)}\right)^{2-\varphi},\quad \varphi=\frac{1+\sqrt{5}}{2}. $$ The golden-ratio exponent of Golomb's sequence survives, while divisor weighting changes the leading constant. We prove the asymptotic without assuming regular variation, by a contraction of upper and lower power envelopes. The argument applies to every nonnegative integer Dirichlet-convolution kernel $w$ with $w(1)=1$ and $\sum_q w(q)q^{-\varphi}<\infty$. For the divisor kernel, a quantitative version gives relative error $O_\gamma((\log n)^{-\gamma})$ for every $0<\gamma<1$. Although the global growth is smooth, the normalized run lengths retain arithmetic fluctuations. A uniform comparison with the divisor sum $\sum_{d\mid m}d^{1-\varphi}$ transfers them to an explicit random Euler product. We prove that its law is singular continuous with full support $[1,\infty)$, determine all real Mellin moments, obtain the sharp maximal order of the fluctuations, and show that sampling sequence positions instead of block labels produces the size-biased law.

Higher-order multivariable Hermite-Kampé de Fériet polynomials provide a unified construction of umbral representations for canonical catastrophe integrals. Expanding the unfolding exponential in these polynomials and integrating against the leading catastrophe kernel gives convergent moment expansions on suitable decay contours. A universal formal umbral operator then expresses the resulting fields through explicit moment ground states. We develop this construction for the fold, cusp and swallowtail, and relate the Euclidean Pearcey field to quartic scalar partition functions. The distinction between formal and analytic umbral frameworks becomes essential when integration is performed before umbral evaluation. For Euclidean Pearcey, the integrated linearized kernel is a rational function whose action is not specified by the original formal rules. The Analytic Umbral Framework supplies a convergent Mellin pairing for this kernel and recovers the independently obtained bivariate Pearcey representation. This establishes, in this example, an exchange of integration and analytic evaluation and identifies a specific extension beyond the formal framework.

Paul Erdős conjectured that the only powers of two whose ternary expansions contain no digit 2 are 1, 4, and 256. Despite extensive computational verification, the conjecture remains open. We reformulate the problem using generalised base-4 repunits. Any admissible exponent is even, say \(n=2s\), and \[ 2^{2s}=4^s=3R_s(4)+1,\qquad R_s(4)=\frac{4^s-1}{3}. \] Thus \(2^{2s}\) omits the ternary digit 2 if and only if \(R_s(4)\) does. For each depth \(m\), the map \(\Phi_m:s\bmod 3^m\mapsto R_s(4)\bmod 3^m\) is a permutation of the residue classes modulo \(3^m\). Hence every prescribed \(m\)-digit ternary suffix has a unique canonical exponent seed modulo \(3^m\), and all exponents producing that suffix form an arithmetic progression. The lifting identity \[ R_{r+q3^m}(4)\equiv R_r(4)+q3^m\pmod{3^{m+1}} \] gives three lifts at each depth, exactly two of which preserve admissibility. The finite maps assemble into an isometric homeomorphism of the 3-adic integers conjugating \(x \mapsto x+1\) with \(y \mapsto 4y+1\). The unresolved global integer problem remains. No proof of the Erdős conjecture is claimed.

We study tilings of closed surfaces by congruent most curvilinear polygons. By relating such tilings to branched coverings determined by planar diagrams and vertex types, we establish necessary and sufficient conditions for a given closed topological surface to admit a metric of constant curvature supporting a tiling with prescribed combinatorial data and number of tiles. We also characterize minimal tilings, which do not descend through nontrivial unbranched coverings to tilings with fewer tiles, in terms of common block decompositions of the associated permutations. Explicit permutation constructions yield infinite families of minimal tilings with prescribed vertex types, even when the planar diagram and vertex types are fixed. In the hyperbolic setting, these results also give finite-index subgroups of the universal tiling symmetry groups that are maximal among torsion-free subgroups, with the index equal to the number of tiles.

Three neural networks are compared as solvers of nonlinear ordinary differential equations: a multilayer perceptron (MLP), a Kolmogorov-Arnold network (KAN) whose edges are B-splines, and a KAN whose edges are Chebyshev series. Each network is trained only on the residual of the equation and on its initial or boundary conditions. The exact solution, or a numerical solution accurate to $10^{-13}$, is used only afterward to measure the error. Five test problems of differential orders one to four are investigated: the logistic equation, a singular Lane-Emden type initial value problem, a singular boundary value problem, the Falkner-Skan equation with suction, a fourth-order problem. The same training budget and the same choice of five activation functions are allocated to all networks. No numerical failures are encountered in any of the \nRuns{} training runs. For every problem, the best KAN is found to be at least as accurate as the best MLP while \paramsKANmin{} to \paramsKANmax{} parameters are used instead of \paramsMLP{}; the margin ranges from a tie to a factor of three. In the Falkner-Skan setting of an earlier study, the B-spline KAN is shown to be 34 to 220 times more accurate at every published test point. Much smaller MLPs are also found to be competitive, whereas at orders three and four, failures are experienced more frequently by KANs, and a coarse basis is required. The derivatives in the equation are distorted by the usual grid extension of a B-spline KAN; consequently, it is replaced with knot insertion, by which the logistic error is lowered more than a hundredfold, though no improvement is provided at order four.