Related papers: Explicit Burgess Bound for Composite Moduli

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…

We study bounds for correlation sums of automorphic coefficients on $\mathrm{GL}_{3,\mathbb{Q}}$ with trace functions of composite moduli. This is a sequel to our previous works with E. Kowalski and W. Sawin.

In this paper, we give the necessary and sufficient conditions for the boundedness of fractional integral operators on the modulation spaces.

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.

We prove that odd unbounded p-summable Fredholm modules are also bounded p-summable Fredholm modules (this is the odd counterpart of a result of A. Connes for the case of even Fredholm modules).

Let $R$ be a standard graded algebra over a field $k$. We prove an Auslander-Buchsbaum formula for the absolute Castelnuovo-Mumford regularity, extending important cases of previous works of Chardin and R\"omer. For a bounded complex of…

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig's label and in terms of…

For any large prime $q$, $1 \leq x \leq q$ and any real $0 \leq k \leq 1$, we prove an upper bound for the following $2k$-th moment $$\displaystyle \sum_{\substack{\chi \bmod q}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k},$$ where…

It is shown that the formula for the variance of combined series yields surprisingly simple proofs of some well known variance bounds.

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…

A method is provided for computing an upper bound of the complexity of a module over a local ring, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which…

We present upper bounds on certain sums which are related to Artin's primitive root conjecture and are also used in counting ray class characters.

We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.

We prove a double-exponential upper bound on the degree and on the complexity of constructing a Janet basis of a $D$-module. This generalizes a well known bound on the complexity of a Gr\"obner basis of a module over the algebra of…

We give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity.

We improve the previuosly known bound for some vertex Folkman numbers.

We propose an algorithm to find a lower bound for the number of cyclic codes over any finite field with any given exponent. Besides, we give a formula to find the exponent of BCH codes.

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give…

Lower bounds for some explicit decision problems over the complex numbers are given.

We give an arithmetic proof of rigidity for postcritically finite polynomials.