International Journal of Number Theory

Let S be a scheme such that 2 is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over S, how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over S and isomorphism classes of pairs $({𝒞},{ℳ})$ where ${𝒞}$ is a quadratic algebra over S and ${ℳ}$ is an invertible ${𝒞}$ -module. Unexpectedly, examples suggest that a refinement of Wood’s bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over S; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples.

Griffin, the second author, and Molnar studied coefficient duality for canonical bases for a broad range of spaces of weakly holomorphic modular forms, showing that the Fourier coefficients of canonical basis elements appear as negatives of Fourier coefficients for elements of a canonical basis of a related space of forms. We investigate the effect of the trace operator on this duality for modular forms for ${\mathrm{\Gamma }}_0(N)$ of genus zero and show exactly when duality still holds after applying the trace operator.

In this paper, we express special values of the L-functions of certain CM elliptic curves which are related to Fermat curves in terms of the special values of generalized hypergeometric functions by comparing Bloch’s element with Ross’s element in the motivic cohomology group.

Bessenrodt and Ono, Chen, Wang and Jia, DeSalvo and Pak were the first to discover the log-subadditivity, the log-concavity and the third-order Turán inequality of the partition function $p(n)$, respectively. Many other important partition statistics have been proved to enjoy similar properties. This paper focuses on the partition function ${\overline{p}}_k(n)$, which counts the number of overpartitions of n with no parts divisible by k. We provide a combinatorial proof to establish that for any $k\ge 3$, the partition function ${\overline{p}}_k(n)$ exhibits the strict log-subadditivity. Specifically, we show that ${\overline{p}}_k(a){\overline{p}}_k(b)>{\overline{p}}_k(a+b)$ for integers $a\ge b\ge 1$ and $a+b\ge k$. Furthermore, we investigate the log-concavity and the third-order Turán inequality for ${\overline{p}}_k(n)$, where $2\le k\le 9$.

Let $r_s(n)$ denote the number of representations of n as a sum of s squares. Hurwitz presented eleven cases in which the generating function of $r_s(an+b)$ is a simple infinite product. In 2004, Cooper and Hirschhorn proved that the generating functions of some infinite families of arithmetic sequences in $r_3(n)$ can be expressed as linear combinations of two given generalized eta-quotients. In this paper, by utilizing some q-series manipulations, we prove that for any $k\ge 0$ and $2\le t\le 50$, the generating functions ${\sum }_{n=0}^{\infty }{r}_{2t-1}(5^{2k+1}n)q^n$, ${\sum }_{n=0}^{\infty }{r}_{2t-1}(5^{2k+2}n)q^n$ and ${\sum }_{n=0}^{\infty }{r}_{2t}(5^{k+1}n)q^n$ can be expressed as linear combinations of certain given generalized eta-quotients. Further, for any $s\ge 3$, we conjecture that these phenomena always exist for $r_s(n)$.

In this paper, by applying some recurrences and iterative computations of certain matrices, we prove some internal congruences modulo powers of 5, 7 and 13 for certain partition functions, including $\ell$-regular partition functions, $\ell$-regular bipartition functions and cubic partition pair function. Further, we conjecture that these internal congruences are just some initial cases in the corresponding internal congruence families. As applications, we establish some non-standard congruences satisfied by the above partition functions. Finally, we raise two related questions that merit further investigation.

In this paper, we consider the relationship between the zeros of degree zero functions from the extended Selberg class and their derivatives left of the critical line. Specifically, we prove Speiser’s result for such functions, that is that the function satisfies the Riemann Hypothesis if and only if its derivative has no zeros left of the critical line. This result allows us to give an easy to check sufficient condition for a degree zero function from the extended Selberg class to satisfy the Riemann hypothesis based on its coefficients. Additionally, we investigate families of such functions parameterized by a real variable and give conditions under which their zero trajectories leave the critical line.

We consider the nonlinear diophantine equation $c_1{p}_1^d+c_2{p}_2^d+\cdots +c_s{p}_s^d=0$ with $c_1,c_2,\dots ,c_s\in \mathbb{Z}$ being non-zero and satisfying $c_1+c_2+\cdots +c_s=0$. We show that for $s\ge 2\lfloor \frac{d^2}{2}\rfloor +1$ and $c\in (1,1+c_0(d,s))$, if the equation has only K-trivial solutions in a subset $\mathcal{𝒜}$ of Piatetski-Shapiro primes up to x corresponding to c, then $|\mathcal{𝒜}|\ll \frac{x^{\frac{1}{c}}}{logx}$ ${(loglogloglogx)}^{\frac{2-s}{dc}+𝜀}$.

Let f be a normalized Maass cusp form for $S{L}_2(\mathbb{Z})$. For $A\ge 2$, we define $M(A)$ as the infimum of all numbers M $(\ge 1)$ such that

Let F be a number field and $p\ge 7$ a rational prime. We obtain a simple descent criterion characterizing those projective Galois representations $\overline{\rho }:G_F\to {\mathrm{\text{PGL}}}_2({𝔽}_p)$ for which the corresponding twist $X_{\overline{\rho }}(p)$ of the principal modular curve of level p is defined over $\mathbb{Q}$. We also give a more concrete version of this result for representations which arise from elliptic curves over cyclic number fields.

We prove the existence of transcendental entire functions f having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and in addition having a prescribed number of k-periodic algebraic orbits, for all $k\ge 1$. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.

There have been results on uniform distribution modulo 1 of sequences of the form ${\{\alpha f(n)\}}_{n=1}^{\infty }$ where $f(n)$ is an arithmetic function and $\alpha$ is an irrational number. For example, ${\{\alpha n\}}_{n=1}^{\infty }$ (Bohl, Sierpiński and Weyl) and ${\{\alpha \mathrm{\Omega }(n)\}}_{n=1}^{\infty }$ (Erdős and Delange) have been shown to be uniformly distributed modulo 1 for all irrational numbers $\alpha$. De Koninck and Kátai have shown that ${\{\alpha \varphi (n)\}}_{n=1}^{\infty }$ and ${\{\alpha \sigma (n)\}}_{n=1}^{\infty }$ are uniformly distributed modulo 1 for a subset of irrational numbers $\alpha$. In this article, we will extend their result by showing that the sequences ${\{\alpha \varphi (n)\}}_{n=1}^{\infty }$ and ${\{\alpha \sigma (n)\}}_{n=1}^{\infty }$ are uniformly distributed modulo 1 when $\alpha$ is a non-Liouville number. The proof will use Weyl’s criterion, upper bounds of exponential functions established by Vinogradov and Vaughan, and the notion of a thin set established by Pollack and Vandehey. There are two corollaries that arise from the result of this article: ${\{10^{\alpha \varphi (n)}\}}_{n=1}^{\infty }$ and ${\{10^{\alpha \sigma (n)}\}}_{n=1}^{\infty }$ are strong Benford sequences for all non-Liouville numbers $\alpha$, and the sequences ${\{F(n)+\alpha \varphi (n)\}}_{n=1}^{\infty }$ and ${\{F(n)+\alpha \sigma (n)\}}_{n=1}^{\infty }$ are uniformly distributed modulo 1 for all non-Liouville numbers $\alpha$ and additive function F.