International Journal of Number Theory

We construct anticyclotomic $p$-adic $L$-functions for ordinary Hilbert modular forms and a formula relating the first derivative of this $p$-adic $L$-function at the central point and Teieltbaum’s ${ℒ}$-invariant defined by Chida, Mok and Park. Our results extend previous works of Bertolini, Darmon, Iovita and Speiss to Hilbert modular forms of parallel weight $2$.

In this paper, we consider an extension of Jacobi’s symbol, the so-called rational $2^k$th power residue symbol. In Sec. 3, we prove a novel generalization of Zolotarev’s lemma. In Secs. 4–6, we show that several hard computational problems are polynomial-time reducible to computing these residue symbols, such as getting nontrivial information about factors of semiprime numbers. We also derive criteria concerning the Quadratic Residuosity Problem.

In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.

Let $n\ge 5$ be an odd integer. It is shown that $\{1^{\sigma (1)},\dots ,n^{\sigma (n)}\}$ is a complete residue system modulo $n$ for some permutation $\sigma$ of $\{1,\dots ,n\}$ if and only if $\frac{1}{2}(n-1)$ is a Sophie Germain prime. Partial results are obtained also for the case $n$ even.

Let $A={(a_0,a_1,\dots ,a_{k-1})}_{\bar{r}}=(\underset{r_0}{\underbrace{a_0,\dots ,a_0}},\underset{r_1}{\underbrace{a_1,\dots ,a_1}},\dots ,\underset{r_{k-1}}{\underbrace{a_{k-1},\dots ,a_{k-1}}})$ be a finite sequence of integers, where $a_0<a_1<\cdots <a_{k-1}$ and $\bar{r}=(r_0,r_1,\dots ,r_{k-1})$ with $r_i\ge 1$ for $i=0,1,\dots ,k-1$. A subsequence sum of $A$ is the sum of all terms of a nonempty subsequence of $A$. Denoted by $S(A)$ the set of all subsequence sums of $A$. In this paper, for given $\bar{r}=(r_0,r_1,\dots ,r_{k-1})$, we give the lower bound for $|S(A)|$ with in terms of $\bar{r}$ and the numbers of positive, negative integers in $A$. We also determine the structure of the finite sequence $A$ of integers for which $|S(A)|$ is minimal. This generalizes the results of Mistri, Pandey and Prakash. Moreover, we give a correction to a result of their paper.

New families of even non-congruent numbers with arbitrarily many distinct prime factors are presented. Methods for generating even non-congruent numbers from existing non-congruent numbers are also given. These results yield new even non-congruent numbers whose factorizations contain primes in all four odd congruence classes modulo eight.

In this note, we investigate the $p$-degree function of an elliptic curve $E/{\mathbb{Q}}_p$. The $p$-degree measures the least complexity of a non-zero $p$-torsion point on $E$. We prove some properties of this function and compute it explicitly in some special cases.

The purpose of the paper is to provide a unified way to formulate zero-sum invariants. Let $G$ be a finite additive abelian group. Let ${ℬ}(G)$ denote the monoid of all zero-sum sequences over $G$. For $\mathrm{\Omega }\subset {ℬ}(G)$, let $d_{\mathrm{\Omega }}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ of length $\left|S\right|\ge t$ has a subsequence in $\mathrm{\Omega }$. We provide some first results and open problems on $d_{\mathrm{\Omega }}(G)$.

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy the same topological property consisting of possessing zeros arbitrarily near every vertical line contained in some substrips inside their critical strip. We also show that the real projections of the zeros of the partial sums of the alternating zeta function, for some particular cases, are dense in their critical intervals.

In 1987, Greene introduced the notion of the finite field analogue of hypergeometric series. In this paper, we give a finite field analogue of Appell series and obtain some transformation and reduction formulas. We also establish the generating functions for Appell series over finite fields.

Let $C$ be a smooth projective algebraic curve of genus $g$, over the finite field $F_q$. A classical result of H. Martens states that the Brill–Noether locus of line bundles ${ℒ}$ in $Pi{c}^{d}C$ with $deg{ℒ}=d$ and $h^0(C,{ℒ})\ge i$ is of dimension at most $d-2i+2$, under conditions that hold when such an ${ℒ}$ is both effective and special. We show that the number of such ${ℒ}$ that are rational over $F_q$ is bounded above by $K_g{q}^{d-2i+2}$, with an explicit constant $K_g$ that grows exponentially with $g$. Our proof uses the Weil estimates for function fields, and is independent of Martens’ theorem. We apply this bound to give a precise lower bound of the form $1-K_g^{\prime }/q$ for the probability that a line bundle in $Pi{c}^{g+1}C(F_q)$ is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree $g+1$ is base point free. This is applicable to the author’s work on fast Jacobian group arithmetic for typical divisors on curves.

In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients $1,2,3,4$ and $6$. We obtain these formulas by constructing explicit bases of the space of modular forms of weight $4$ on ${\mathrm{\Gamma }}_0(48)$ with character.

We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for ${\mathrm{\Gamma }}_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the distribution of the signs of ${\{a_f(p^m)\}}_p$, where $p$ runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power $L$-functions.

We give a list of ${\mathrm{\text{PGL}}}_2({𝔽}_{\ell })$ number fields for $\ell \ge 11$ which have rational companion forms. Our list has 53 fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.

Let $f$ be a fixed self-dual Hecke–Maass cusp form for $\mathrm{\text{SL}}(3,\mathbb{Z})$ and $\{{\mu }_j\}$ be an orthogonal basis of odd Hecke–Maass cusp forms for $\mathrm{\text{SL}}(2,\mathbb{Z})$. We prove an asymptotic formula for the average of the first derivative of the Rankin–Selberg $L$-function of $f$ and ${\mu }_j$ at the center point $s=\frac{1}{2}$. This implies the non-vanishing results for the first derivative of these $L$-functions at the center point $s=\frac{1}{2}$.

In this paper, we study products of consecutive integers and prove an optimal bound on their greatest common factors. In contrast, we obtain a modest upper bound for the greatest common factor in the perfect square situation. Using this and an upper bound on the size of solutions to hyperelliptic curves, we prove a gap principle when $a^2(a^2+1)$ divides $b^2(b^2+1)$ with some additional restrictions. We also obtain a stronger gap principle under the abc conjecture.

In this note, we study conjectures of Artin and Langlands and derive the automorphy of all solvable groups of order at most 200, three groups excepted.

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.