On the divisibility among power GCD and power LCM matrices on gcd-closed sets

Let [Formula: see text] and [Formula: see text] be positive integers and let [Formula: see text] be a set of [Formula: see text] distinct positive integers. For [Formula: see text], one defines [Formula: see text]. We denote by [Formula: see text] (respectively, [Formula: see text]) the [Formula: see text] matrix having the [Formula: see text]th power of the greatest common divisor (respectively, the least common multiple) of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry. In this paper, we show that for arbitrary positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], the [Formula: see text]th power matrices [Formula: see text] and [Formula: see text] are both divisible by the [Formula: see text]th power matrix [Formula: see text] if [Formula: see text] is a gcd-closed set (i.e. [Formula: see text] for all integers [Formula: see text] and [Formula: see text] with [Formula: see text]) such that [Formula: see text]. This confirms two conjectures of Shaofang Hong proposed in 2008.

Rationality of the Petersson Inner Product of Cohen’s Kernels

By explicitly calculating and then analytically continuing the Fourier expansion of the twisted double Eisenstein series [Formula: see text] of Diamantis and O’Sullivan, we prove a formula of the Petersson inner product of Cohen’s kernel and one of its twists, and obtain a rationality result. This extends a result of Kohnen and Zagier.

Khintchine’s theorem with rationals coming from neighborhoods in different places

The Duffin–Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine’s theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine’s theorem studied. We prove versions of Khintchine’s theorem where the rational numbers are sourced from a ball in some completion of [Formula: see text] (i.e. Euclidean or [Formula: see text]-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarník’s theorem.

Some q-supercongruences related to Swisher’s (H.3) conjecture

We first give a [Formula: see text]-analogue of a supercongruence of Sun, which is a generalization of Van Hamme’s (H.2) supercongruence for any prime [Formula: see text]. We also give a further generalization of this [Formula: see text]-supercongruence, which may also be considered as a generalization of a [Formula: see text]-supercongruence recently conjectured by the second author and Zudilin. Then, by combining these two [Formula: see text]-supercongruences, we obtain [Formula: see text]-analogues of the following two results: for any integer [Formula: see text] and prime [Formula: see text] with [Formula: see text] [Formula: see text] [Formula: see text] which are generalizations of Swisher’s (H.3) conjecture modulo [Formula: see text] for [Formula: see text]. The key ingredients in our proof are the ‘creative microscoping’ method, the [Formula: see text]-Dixon sum, Watson’s terminating [Formula: see text] transformation, and properties of the [Formula: see text]-adic Gamma function.

Exceptional sets related to the largest digits in Lüroth expansions

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

A sharp estimate of the discrepancy of a concatenation sequence of inversive pseudorandom numbers with consecutive primes

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.

Remarks on the rightmost critical value of the triple product L-function

Let [Formula: see text] be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product [Formula: see text]-functions [Formula: see text], where [Formula: see text] and [Formula: see text] run over an orthogonal basis of [Formula: see text] consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product [Formula: see text]-functions.

Galois covers of ℙ1 and number fields with large class groups

For each finite subgroup [Formula: see text] of [Formula: see text], and for each integer [Formula: see text] coprime to [Formula: see text], we construct explicitly infinitely many Galois extensions of [Formula: see text] with group [Formula: see text] and whose ideal class group has [Formula: see text]-rank at least [Formula: see text]. This gives new [Formula: see text]-rank records for class groups of number fields.

Norm form equations and linear divisibility sequences

Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be written as tuples of linear recurrence sequences. We show that for certain families of norm forms defined over quartic fields, there exist integrally equivalent forms making any one fixed coordinate sequence a linear divisibility sequence.

Twisted moments of GL(3) ×GL(2) L-functions

We compute an asymptotic formula for the twisted moment of [Formula: see text] [Formula: see text]-functions and their derivatives. As an application, we prove that symmetric-square lifts of [Formula: see text] Maass forms are uniquely determined by the central values of the derivatives of [Formula: see text] [Formula: see text]-functions.