International Journal of Number Theory

Many papers have been recently devoted to the study of the radial behavior as $z\to 1^{-}$ of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like $(1+o(1))C(z)/{(1-z)}^{\rho }$ for some $\rho \in ℂ$ and $C(z)$ is a real analytic function of $z\in (0,1)$ such that $C(z)=C(z^r)$. They did not provide a formula for $C(z)$ which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of $C(z)$ as an exponential of a Fourier series in the variable $loglog(1/z)/log(r)$ for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute $C(z)$ associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in $\mathbb{Q}[[z]]$), we provide an explicit expression of $C(z)$ for every r-Mahler function of order 2 under mild assumptions on the coefficients in $ℝ(z)$ of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1.

Gaussian periods have been studied for centuries in the realms of number theory, field theory, cryptography, and elsewhere. However, it was only within the last decade or so that they began to be studied from a visual perspective. By plotting Gaussian periods in the complex plane, various interesting and insightful patterns emerge, leading to various conjectures and theorems about their properties.

In this paper, we offer a description of Gaussian periods, along with examples of the structure that can occur when plotting them in the complex plane. In addition to this, we offer two ways in which this study can be generalized to other situations — one relating to supercharacter theory, the other relating to class field theory — along with discussions and visual examples of each.

Let p be a prime number. For each natural number n, we study the behavior of the function $f_p(n)$ which enumerates the number of factorizations $ab=n$ with $a+b$ a perfect square (mod p). The study of this function is inspired by the cognate function $f(n)$ which enumerates the number of factorizations $ab=n$ with $a+b$ a perfect square. The descent theory of elliptic curves would show that if $f(n)$ is unbounded for squarefree values of n, then there are elliptic curves over the rational number field with arbitrarily large rank. In this note, we show for every prime p, $f_p(n)$ is unbounded as n ranges over squarefree values, thus providing some evidence for the conjecture that $f(n)$ is unbounded for squarefree n.

The Pólya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of consecutiveness of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large Pólya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields.

Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan–Gross–Prasad conjecture via the Gross–Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross–Zagier formula.

For a prime p, positive integers $r,n$, and a polynomial f with coefficients in ${𝔽}_{p^r}$, let $W_{p,r,n}(f)=f^n\left({𝔽}_{p^r}\right)\backslash f^{n+1}\left({𝔽}_{p^r}\right)$. As n varies, the $W_{p,r,n}(f)$ partition the set of strictly preperiodic points of the dynamical system induced by the action of f on ${𝔽}_{p^r}$. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of ${𝔽}_{p^r}$ lying in a given $W_{p,r,n}(f)$. Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both upper and lower bounds for the resulting averages.

Let p be an odd prime and ${𝔽}_p$ be the finite field with p elements. This paper focuses on the study of the values of a generic family of hypergeometric functions in the p-adic setting, which we denote by ${}_{3n-1}{G}_{3n-1}(p,t)$, where $n\ge 1$ and $t\in {𝔽}_p$. These values are expressed in terms of numbers of zeros of certain polynomials over ${𝔽}_p$. These results lead to certain p-adic analogues of classical hypergeometric identities. Namely, we obtain p-adic analogues of particular cases of a Gauss’s theorem and a Kummer’s theorem. Moreover, we examine the zeros of these functions. For example, if n is odd, we characterize t for which ${}_{3n-1}{G}_{3n-1}(p,t)$ has zeros. In contrast, we show that if n is even, then the function ${}_{3n-1}{G}_{3n-1}(p,t)$ has no zeros for any prime p apart from the trivial case when $t=0$.

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall’s identity for the generalized hypergeometric function ${}_5{F}_4$. Similar results for interpolated refined symmetric multiple zeta values and half refined symmetric multiple zeta values are also discussed.

Let F be a field of formal series over a finite field and ${\mathcal{ℬ}}_d$ be the affine building associated to ${\mathrm{\text{PGL}}}_d(F)$. Given a lattice $\mathrm{\Gamma }$ in ${\mathrm{\text{PGL}}}_d(F)$, the complex arising as a quotient $\mathrm{\Gamma }\backslash {\mathcal{ℬ}}_d$ is called weakly Ramanujan if every nontrivial discrete simultaneous spectrum of the colored adjacency operators $A_1,A_2,\dots ,A_{d-1}$ acting on $L^2(\mathrm{\Gamma }\backslash {\mathcal{ℬ}}_d)$ is contained in the simultaneous spectrum of those operators acting on $L^2({\mathcal{ℬ}}_d)$. In this paper, we prove that the standard non-uniform arithmetic quotient ${\mathrm{\text{PGL}}}_4({𝔽}_q[t])\backslash {\mathcal{ℬ}}_4$ of ${\mathrm{\text{PGL}}}_4(F)$ is weakly Ramanujan.

Mock theta functions were first introduced by Ramanujan in his last paper to Hardy. Moreover, some other mock theta functions were presented in his lost notebook. It is well known that all the classical mock theta functions can be expressed by the universal mock theta functions $g_2(x,q)$ and $g_3(x,q)$. In this paper, we establish some generalized mock theta functions and express them in terms of Appell–Lerch sums. Meanwhile, we show that some of Ramanujan’s two-parameter mock theta functions are the special cases of these functions.

Recently, establishing the Hecke-type double sums forms for Eulerian series has received a great deal of attention to seek special cases of q-hypergeometric series. In this paper, with the aid of some basic hypergeometric series identities given by Chen and Wang, and Liu, we discover many new Eulerian series and express them in terms of Hecke-type double sums and Appell–Lerch sums. Based on the modern sense for the mock theta functions, we find that some of these series are mock theta functions and the others are mixed mock theta functions. Furthermore, from the viewpoint of Hecke-type double sums, we find three string function identities with levels 2, 4, and 6.