The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note, we observe that the sequence of run lengths in such a sequence, as well as the starting and ending positions of the nth run, is $2$-synchronised and hence computable by a finite automaton. As a specific consequence, we obtain the recent results of Bunder, Bates and Arnold [‘The summed paperfolding sequence’, Bull. Aust. Math. Soc. 110 (2024), 189–198] in much more generality, via a different approach. We also prove results about the critical exponent and subword complexity of these run-length sequences.

Let $f $ be a normalized Hecke eigenform of even weight $k \geq 2$ for $SL_2(\mathbb {Z})$. In this article, we establish an asymptotic formula for the shifted convolution sum of a general divisor function, where the sum involves the Fourier coefficients of a multi-folded L-function weighted with a kernel function.

For any integers x and y, let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of x and y, respectively. Let $a,b$ and n be positive integers, and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. We denote by $(S^a)$ and $[S^a]$ the $n\times n$ matrices having the ath power of $(x_i,x_j)$ and $[x_i,x_j]$, respectively, as the $(i,j)$-entry. Bourque and Ligh [‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 65–74] showed that if S is factor closed (that is, S contains all positive divisors of any element of S), then the GCD matrix $(S)$ divides the LCM matrix $[S]$ (written as $(S)\mid [S]$) in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008] proved that $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$ when $a\mid b$ and S is a divisor chain (namely, there is a permutation $\sigma $ of order n such that $x_{\sigma (1)}\mid \cdots \mid x_{\sigma (n)}$). In this paper, we show that if $a\mid b$ and S is factor closed, then $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$. The proof is algebraic and p-adic. Our result extends the Bourque–Ligh theorem. Finally, several interesting conjectures are proposed.

We show that on a $\sigma $-finite measure-preserving system $X = (X,\nu , T)$, the non-conventional ergodic averages

$$ \begin{align*} \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x) \end{align*} $$

$f \in L^{p_1}(X)$

$g \in L^{p_2}(X)$

$1/p_1 + 1/p_2 \leq 1$

$2$

$\Lambda $

$1$

$\mu $

We strengthen known results on Diophantine approximation with restricted denominators by presenting a new quantitative Schmidt-type theorem that applies to denominators growing much more slowly than in previous works. In particular, we can handle sequences of denominators with polynomial growth and Rajchmann measures exhibiting arbitrary slow decay, allowing several applications. For instance, our results yield non-trivial lower bounds on the Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers (each with restricted denominators) and enable the construction of Salem subsets of well-approximable numbers of arbitrary Hausdorff dimension.

We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb {Z}/2\mathbb {Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a canonical Jacobian elliptic fibration and the modular form generators appear as coefficients in the Weierstrass-type equations describing these fibrations.

We prove that the structure $(\mathbb {Z},<,+,R)$ is distal for all congruence-periodic sparse predicates $R\subseteq \mathbb {N}$. We do so by constructing a strong honest definition for every formula $\phi (x;y)$ with $\lvert {x}\rvert =1$, providing a rare example of concrete distal decompositions.

Determining the polynomials $D \in {\mathbb Z}[x]$ such that the polynomial Pell equation ${P^2-DQ^2=1}$ has nontrivial solutions $P,Q$ in ${\mathbb Q}[x]$ (and in ${\mathbb Z}[x]$) is an open question. In this article, we consider the generalized polynomial Pell equation $P^2-DQ^2=n$, where $D \in {\mathbb Z}[x]$ is a monic quadratic polynomial and n is a nonzero integer. For $n=1$, such an equation always has nontrivial solutions in ${\mathbb Q}[x]$, but for a non-square integer n, the generalized polynomial Pell equation $P^2-DQ^2=n$ may not always have a solution in ${\mathbb Q}[x]$. Depending on n, we determine the polynomials $D=x^2+cx+d$, for which the equation $P^2-DQ^2=n$ has nontrivial solutions in ${\mathbb Q}[x]$ and in ${\mathbb Z}[x]$. Taking $n=-1$, this allows us to solve the negative polynomial Pell equation completely for any such D. An interesting feature is that there are certain polynomials D for which the generalized polynomial Pell equation has nontrivial solutions in ${\mathbb Z}[x]$, but only finitely many, whereas the solutions in ${\mathbb Q}[x]$ are infinitely many. Finally, we determine the monic quadratic polynomials D for which the solutions of $P^2-DQ^2=n$ in ${\mathbb Z}[x]$ exhibit this finiteness phenomenon.

Pursuing ideas in [6], we determine the isometry classes of unimodular lattices of rank $28$, as well as the isometry classes of unimodular lattices of rank $29$ without nonzero vectors of norm $\leq 2$. We also provide some invariant that allows to distinguish these lattices and to independently check with a computer that our lists are complete.

We extend the work of N. Zubrilina on murmuration of modular forms to the case when prime-indexed coefficients are replaced by squares of primes. Our key observation is that the shape of the murmuration density is the same.

Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and conjecture a second congruence family by powers of 7, which may be amenable to similar techniques.

Let $a(n)$ be the nth Dirichlet coefficient of the automorphic L-function or the Rankin–Selberg L-function. We investigate the cancellation of $a(n)$ over sequences linked to the Waring–Goldbach problem, by establishing a non-trivial bound for the additive twisted sums over primes on ${\mathrm {GL}}_m$. The bound does not depend on the generalized Ramanujan conjecture or the non-existence of Landau–Siegel zeros. Furthermore, we present an application associated with the Sato–Tate conjecture and propose a conjecture about the Goldbach conjecture on average bound.

In this paper, we generalise to the family of Fermat quartics $X^4 + Y^4 = 2^m, m \in \mathbb {Z}$, a result of Aigner [‘Über die Möglichkeit von $x^4 + y^4 = z^4$ in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely $\mathbb {Q}(\sqrt {-7})$, that contains solutions to the Fermat quartic $X^4 + Y^4 = 1$. The $m \equiv 0 \pmod 4$ case is due to Aigner. The $m \equiv 2 \pmod 4$ case follows from a result of Emory [‘The Diophantine equation $X^4 + Y^4 = D^2Z^4$ in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases $m \equiv 1, 3 \pmod 4$, classifying for $m \equiv 1 \pmod 4$ the infinitely many quadratic number fields that contain solutions, and proving for $m \equiv 3 \pmod 4$ that $\mathbb {Q}(\sqrt {2})$ and $\mathbb {Q}(\sqrt {-2})$ are the only quadratic number fields that contain solutions.

Martin, Mossinghoff, and Trudgian [19] recently introduced a family of arithmetic functions called “fake $\mu $’s,” which are multiplicative functions for which there is a $\{-1,0,1\}$-valued sequence $(\varepsilon _j)_{j=1}^{\infty }$ such that $f(p^j) = \varepsilon _j$ for all primes p. They investigated comparative number-theoretic results for fake $\mu $’s and, in particular, proved oscillation results at scale $\sqrt {x}$ for the summatory functions of fake $\mu $’s with $\varepsilon _1=-1$ and $\varepsilon _2=1$. In this article, we establish new oscillation results for the summatory functions of all nontrivial fake $\mu $’s at scales $x^{1/2\ell }$ where $\ell $ is a positive integer (the “critical index”) depending on f; for $\ell =1$ this recovers the oscillation results in [19]. Our work also recovers results on the indicator functions of powerfree and powerfull numbers; we generalize techniques applied to each of these examples to extend to all fake $\mu $’s.

We prove the central limit theorem (CLT), the first-order Edgeworth expansion and a mixing local central limit theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb {R}$. The class of observables in the CLT and the MLCLT on $\mathbb {R}$ include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber [Sampling the Lindelöf hypothesis with the Cauchy random walk. Proc. Lond. Math. Soc. (3) 98 (2009), 241–270] and Steuding [Sampling the Lindelöf hypothesis with an ergodic transformation. RIMS Kôkyûroku Bessatsu B34 (2012), 361–381] who have proven the strong law of large numbers for sampling the Lindelöf hypothesis.

For every positive integer n, we introduce a set ${\mathcal {T}}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb {Z}^2\to {\mathcal {T}}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n\geq 1$, we show that the Wang shift ${\Omega }_n$, defined as the set of valid configurations over the tiles ${\mathcal {T}}_n$, is self-similar, aperiodic and minimal for the shift action. We say that $\{{\Omega }_n\}_{n\geq 1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $\Omega _n$ is the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the n-th metallic mean, and in particular, the golden mean when $n=1$, and the silver mean when $n=2$. When $n=1$, the set of Wang tiles ${\mathcal {T}}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles.

Granville–Soundararajan, Harper–Nikeghbali–Radziwiłł and Heap–Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment.

Let $\ell $ be an odd prime. We investigate the enumeration of cyclic extensions of degree $\ell $ over $\mathbb {Q}$ subject to specified local conditions. By ordering these extensions according to their conductors, we derive an asymptotic count with a power-saving error term. As a consequence of our results, we analyze the distribution of values of L-functions associated with these extensions in the critical strip.

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erdős—Hooley $\Delta$-function, we derive lower bounds for the cardinality of those integers not exceeding a given limit that are expressible as certain sums of powers.

We study the freeness problem for multiplicative subgroups of $\operatorname{SL}_2(\mathbb{Q})$. For $q = r/p$ in $\mathbb{Q} \cap (0,4)$, where p is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $\Delta_q$ generated by

\[A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} \text{ and } Q_q = \begin{pmatrix}1 & q \\ 0 & 1\end{pmatrix}.\]

$\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$r \leq 4$

$r = 5$

$r \in \{ p-1, p+1, (p+1)/2 \}$

$\Delta_{r/p}$

$J_2(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$J_2(r)$