Bilinear forms in Weyl sums for modular square roots and applications

For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of $g_{\mathbb Z}(n)$ squares of integral linear forms. We show that as $n$ goes to infinity, the growth of $g_{\mathbb Z}(n)$ is at most an exponential of $\sqrt{n}$. Our result improves the best known upper bound on $g_{\mathbb Z}(n)$ which is in the...

We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ``arithmetic Bertini theorem'' conjectured by Poonen for $\mathbb{P}^1_\mathbb{Z}$. Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number field...

In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at $0,$ and arithmetic functions. We will also give a relation between the s...

In these notes, we refine Mitsui's Prime Number Theorem from 1957, which for a number field $K$ predicts how many prime elements there are in bounded convex sets in $K \otimes_{\mathbf Q} \mathbf R$, by incorporating potential Siegel zeros of Hecke L-functions. This allows the norm of the modulus $\mathbf N (\mathfrak q) $ to grow at a pseudopolynomial rate with respect to the size $X$ of the convex set as opposed to powers of $\log X$. The extra flexibility and precision wil...

This paper deals with function field analogues of the famous theorem of Landau which gives the asymptotic density of sums of two squares in $\mathbb{Z}$. We define the analogue of a sum of two squares in $\mathbb{F}_q[T]$ and estimate the number $B_q(n)$ of such polynomials of degree $n$ in two cases. The first case is when $q$ is large and $n$ fixed and the second case is when $n$ is large and $q$ is fixed. Although the methods used and main terms computed in each of the t...

In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k_{0}, using the values of the zeta function \zeta_{k_{0}} at negative integers as our ``higher Bernoulli numbers''. In the case where k_{0} is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The auth...

We prove a uniform estimate for sums of Hecke--Maass eigenvalues squared over primes in short intervals that can be regarded as an analogue of Hoheisel's classical prime number theorem for all real analytic cusp forms. Our argument is modelled after our treatment of Linnik's least prime number theorem for arithmetic progressions (Tata LN 72) and depends on recent works in the theory of automorphic representations. We stress that constants in the present work, including those ...

Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in the large $q$ limit, finding a connection to the $z$-measures first investigated in the context of harmonic analysis on the infinite symmetric group. A similar connection to $z$-measures is established for sums over short intervals of the div...

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$. Further, a method is described which computes representations of primes from reduced quadratic forms by means of the integral roots of polynomials over $\mathbb Z$. Lastly, some progress...

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.