相关论文: Uniform bounds for norms of theta series and arith…

Let $s\geq 8$ be an integer and $P$ be a set of primes with relative lower density greater than $\sqrt{1-\min\{s,16\}/32}$. We prove that every sufficiently large integer $n\equiv s({\rm mod}24)$ can be represented by a sum of $s$ squares…

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the…

Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of…

In this paper, we investigate the size of moments of quadratic character sums averaged over the family of fundamental discriminants. We obtain an asymptotic formula for all integer moments in a restricted range of parameters using a…

We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19…

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants…

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq…

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the…

We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier…

In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…

Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\rm GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations.…

We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of…

Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{\nu n}$ with $\nu>1/2$.

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+\delta}$ with a…

Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best…

Norm forms, examples of which include $x^2 + y^2$, $x^2 + x y - 57 y^2$, and $x^3 + 2 y^3 + 4 z^3 - 6 x y z$, are integral forms arising from norms on number fields. We prove that the natural density of the set of integers represented by a…

We show that for every fixed $A>0$ and $\theta>0$ there is a $\vartheta=\vartheta(A,\theta)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log…

Let $p_n$ be the maximal sum of the entries of $A^2$, where $A$ is a square matrix of size $n$, consisting of the numbers $1,2,\ldots,n^2$, each appearing exactly once. We prove that $m_n=\Theta(n^7)$. More precisely, we show that…