Narrow Results

A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a POVM in ${ℂ}^K$ consisting of $K^2$ positive operators of rank one such that all of whose Hermite inner products are equal. SIC-POVMs are important in quantum information theory, which have many applications in quantum state tomography, quantum cryptography and basic research in quantum mechanics. However, it is very difficult to construct SIC-POVMs. Therefore, many scholars have focused on approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) for which the Hermite inner products are close to equal. In this paper, two new constructions of ASIC-POVMs are provided by using character sums and some special functions over finite fields.

In this paper, we precise the asymptotic behavior of Newton polygons of L-functions associated to character sums, coming from certain n variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum Δ = Δ₁⊕Δ₂ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behavior of Newton polygons for multivariable polynomials when the generic Newton polygon differs from the combinatorial (Hodge) polygon associated to the polyhedron.

In this work we present an explicit relation between the number of points on a family of algebraic curves over 𝔽_q and sums of values of certain hypergeometric functions over 𝔽_q. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over 𝔽_q in some particular cases. A general conjecture relating these last two is presented and advances toward its proof are shown in the last section.

Let K be a cyclic number field of prime degree ℓ. Heilbronn showed that for a given ℓ there are only finitely many such fields that are norm-Euclidean. In the case of ℓ = 2 all such norm-Euclidean fields have been identified, but for ℓ ≠ 2, little else is known. We give the first upper bounds on the discriminants of such fields when ℓ > 2. Our methods lead to a simple algorithm which allows one to generate a list of candidate norm-Euclidean fields up to a given discriminant, and we provide some computational results.

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface

The Burgess inequality is the best upper bound we have for incomplete character sums of Dirichlet characters. In 2006, Booker gave an explicit estimate for quadratic Dirichlet characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than 10¹⁴⁰. Both of their explicit estimates are on restricted ranges. In this paper, we prove an explicit estimate that works for any range. We also improve McGown's estimates in a slightly narrower range, getting explicit estimates for characters of any order. We apply the estimates to the question of how large must a prime p be to ensure that there is a kth power non-residue less than p^1/6.

We study the character sums

For any odd prime number $p$, let $(\cdot |p)$ be the Legendre symbol, and let $n_1(p)<n_2(p)<\cdots$ be the sequence of positive nonresidues modulo $p$, i.e. $(n_k|p)=-1$ for each $k$. In 1957, Burgess showed that the upper bound $n_1(p){\ll }_{\epsilon }{p}^{{(4\sqrt{e})}^{-1}+\epsilon }$ holds for any fixed $\epsilon >0$. In this paper, we prove that the stronger bound

We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod $p$ less than $p^{1/2+3/(4\sqrt{e})+𝜖}$.

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.

We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the square root cancellation in the context of the parabola, whereas its analogues are wide open conjectures for the circle and the hyperbola. We also obtain essentially sharp upper bounds for the latter lattice points problem associated with the parabola. Our proofs utilize techniques in Fourier analysis, quadratic Gauss sums and character sums.

We prove an explicit version of Burgess’ bound on character sums for composite moduli.

In this paper, we give an explicit upper bound on $h(p)$, the least primitive root modulo $p^2$. Since a primitive root modulo $p$ is not primitive modulo $p^2$ if and only if it belongs to the set of integers less than $p$ which are $p$th power residues modulo $p^2$, we seek the bounds for $N_1(H)$ and $N_2(H)$ to find $H$ which satisfies $N_1(H)-N_2(H)>0$, where, $N_1(H)$ denotes the number of primitive roots modulo $p$ not exceeding $H$, and $N_2(H)$ denotes the number of $p$th powers modulo $p^2$ not exceeding $H$. The method we mainly use is to estimate the character sums contained in the expressions of the $N_1(H)$ and $N_2(H)$ above. Finally, we show that $h(p)<p^{0.74}$ for all primes $p$. This improves the recent result of Kerr et al.

In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields ${𝔽}_p$ for odd prime $p$ and guarantee the existence of a $k$-Diophantine $m$-tuple provided $p$ is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in ${𝔽}_p$ as well as an asymptotic formula for the number of $k$-Diophantine $k$-tuples.

We evaluate the smoothed first moment of central values of a family of quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size $O(X^{1/4+𝜀})$ under the generalized Riemann hypothesis. The same approach also allows us to obtain asymptotic formulas for all $X$, $Y$ for a smoothed double character sum involving ${\sum }_{N(m)\le X,N(n)\le Y}(\frac{m}{n})$, where $(\frac{\cdot }{n})$ denotes the quadratic symbol in the Gaussian field.

In this paper, we study several topics on additive decompositions of primitive elements in finite fields. Also we refine some bounds obtained by Dartyge and Sárközy as well as Shparlinski.

For an odd prime p, let ${𝔽}_p$ be the finite field of p elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any $a,b,c\in {𝔽}_p^{*}$, we also obtain new upper bounds of the following double character sum

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.

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.