相关论文: A lower bound for the variance in arithmetic progr…

We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-\epsilon}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of…

We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial…

We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.

We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower…

We give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and non-periodic…

We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order…

We show that the hereditary discrepancy of homogeneous arithmetic progressions is lower bounded by $n^{1/O(\log \log n)}$. This bound is tight up to the constant in the exponent. Our lower bound goes via proving an exponential lower bound…

In this paper, we provide tight lower bounds for the oracle complexity of minimizing high-order H\"older smooth and uniformly convex functions. Specifically, for a function whose $p^{th}$-order derivatives are H\"older continuous with…

The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may…

We study a general class of multiplicative functions by relating "short averages" to its "long average". More precisely, we estimate asymptotically the variance of such a class of functions in short intervals using Fourier analysis and…

We show that any Algebraic Branching Program (ABP) computing the polynomial $\sum_{i = 1}^n x_i^n$ has at least $\Omega(n^2)$ vertices. This improves upon the lower bound of $\Omega(n\log n)$, which follows from the classical result of Baur…

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >=…

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known…

Let $d_k(n) = \sum_{n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{N} \to \mathbb{C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in…

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0<\alpha<1$, in the vicinity of an exceptional boundary point where all such functions…

Let $f : \mathbf{N} \rightarrow \mathbf{C}$ be a bounded multiplicative function. Let $a$ be a fixed integer (say $a = 1$). Then $f$ is well-distributed on the progression $n \equiv a \pmod{q} \subset \{1,\dots, X\}$, for almost all primes…

We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denumerant function. These bounds are tight enough to get the asymptotic behavior of the denumerant.

Assume that $ y < N$ are integers, and that $ (b,y) =1$. Define an average along the primes in a progression of diameter $ y$, given by integer $ (b,y)=1 $. \begin{align*} A_{N,y,b} := \frac{\phi (y)}{N} \sum _{\substack{n <N\\n\equiv…

Based on the~method of subordinating functions we prove bounds for the minimal error of approximations of $n$-fold convolutions of probability measures by free infinitely divisible probability measures.

We prove two estimates for the Barban--Davenport--Halberstam type variance of a general complex sequence in arithmetic progressions. The proofs are elementary, and our estimates are capable of yielding an asymptotic for the variance when…