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

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

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

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

$r \leq 4$

$r = 5$

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

$\Delta_{r/p}$

$J_2(r)$

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

$J_2(r)$

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

Using an explicit Eichler–Shimura–Harder isomorphism, we establish the analog of Manin’s rationality theorem for Bianchi periods and hence special values of L-functions of Bianchi cusp forms. This gives a new short proof of a result of Hida in the case of Euclidean imaginary quadratic fields. In particular, we give an explicit proof using the space of Bianchi period polynomials constructed by Karabulut and describe the action of Hecke operators.

It is conjectured that for any fixed relatively prime positive integers a,b and c all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers x, y and z, except for specific cases. We develop the methods in our previous work which rely on a variety from Baker’s theory and thoroughly study the conjecture for cases where c is small relative to a or b. Using restrictions derived from the hypothesis that there is more than one solution to the equation, we obtain a number of finiteness results on the conjecture. In particular, we find some, presumably infinitely many, new values of c with the property that for each such c the conjecture holds true except for only finitely many pairs of a and b. Most importantly we prove that if $c=13$ then the equation has at most one solution, except for $(a,b)=(3,10)$ or (10,3) each of which gives exactly two solutions. Further, our study with the help of the Schmidt Subspace Theorem among others more, brings strong contributions to the study of Pillai’s type Diophantine equations, notably a general and satisfactory result on a well-known conjecture of M. Bennett on the equation $a^x-b^y=c$ for any fixed positive integers a,b and c with both a and b greater than 1. Some conditional results are presented under the abc-conjecture as well.

We show that the set of Liouville numbers has a rich set-theoretic structure: it can be partitioned in an explicit way into an uncountable collection of subsets, each of which is dense in the real line. Furthermore, each of these partitioning subsets can be similarly partitioned, and the process can be repeated indefinitely.

We formulate Guo–Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

We show that smooth numbers are equidistributed in arithmetic progressions to moduli of size $x^{66/107-o(1)}$. This overcomes a longstanding barrier of $x^{3/5-o(1)}$ present in previous works of Bombieri, Friedlander and Iwaniec, Fouvry and Tenenbaum, Drappeau, and Maynard. We build on Drappeau’s variation of Linnik’s dispersion method and on exponential sum manipulations of Maynard, ultimately relying on optimized Deshouillers–Iwaniec-type estimates for sums of Kloosterman sums.

We show that each local field $\mathbb{F}_q(\!(t)\!)$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most 1 by (certain small quotients of) its absolute Galois group together with natural axioms concerning the p-torsion of its Brauer group. This complements previous work by Efrat and Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic p.

We will give a precise and explicit asymptotic estimate for the characteristic of the Riemann zeta function $\zeta $ with an error term of order $O(\frac {\log r}{r})$ and a corresponding asymptotic estimate for the number of fixed points of $\zeta $.