International Journal of Number Theory

We study a family of residual crank generating functions defined on overpartitions, the so-called $k$th residual cranks. Specifically, the moment generating functions associated to these cranks exhibit quasimodularity properties which are dependent on the choice of $k$. We also show that the second moments of these cranks admit a combinatoric interpretation as weighted overpartition counts. This interpretation gives a refinement to an existing inequality between crank moments of differing modulus $k$.

There are many results in the literature concerning power values, equal values or more generally, polynomial values of lattice point counting polynomials. In this paper, we prove various finiteness results for polynomial values of polynomials counting the lattice points on the surface of an $n$-dimensional cube, pyramid and simplex.

Drinfeld modules and A-motives s are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and $A$-motives, one can construct their $l$-adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for $A$-motives. Here, we clarify the rank-one case for A-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.

Several new estimates for the $2$-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems, including a long-standing open conjecture by Lengyel. The estimates and criteria all depend on our previous analysis of powers of $2$ in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.

Three transformation formulae are established for the partial sum of Bailey’s well-poised ${}_6{\psi }_6$-series. Their particular cases provide $q$-analogues of Guillera’s two series for ${\pi }^{\pm 2}$ with convergence rate $\frac{27}{64}$, and for other classical $\pi$-related infinite series.

It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of Guo and Xie, some evaluation formulas with even arguments and the restricted sum formulas of Hoffman and their generalizations.

The $\bar{\mathbb{Q}}$-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of $\mathbb{Q}$-rational functions over $\mathbb{Q}$-semi-algebraic domains in ${ℝ}^d$. The Kontsevich–Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stokes formula. In this paper, we prove that every non-zero real period can be represented as the volume of a compact ${ℝ}_{\mathrm{\text{alg}}}$-semi-algebraic set obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich–Zagier period conjecture.

Recurrence formulas for generalized poly-Bernoulli polynomials are given. The formula gives a positive answer to a question raised by Kaneko. Further, as applications, annihilation formulas for Arakawa-Kaneko type zeta-functions and a counting formula for lonesum matrices of a certain type are also discussed.

We prove that the equation ${(x-y)}^4+x^4+{(x+y)}^4=z^n$ has no integer solutions $x,y,z$ with $gcd(x,y)=1$ for all integers $n>1$. We mainly use a modular approach with two Frey $\mathbb{Q}$-curves defined over the field $\mathbb{Q}(\sqrt{30})$.