Related papers: Rational exponential sums over the divisor functio…
We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.
In [arXiv:1504.07804], Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb…
Recently, Bordell\'{e}s, Dai, Heyman, Pan and Shparlinski in \cite{Igor} considered a partial sum involving the Euler totient function and the integer parts $\lfloor x/n\rfloor$ function. Among other things, they obtained reasonably tight…
We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc…
We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve…
Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums…
If the denominator of a rational function of several variables is sum of even powers and the numerator is a monomial, then we give a numerical criterion, using the exponents involved in the expression of the rational function, to decide if…
The well known analytical formula for $SU(2)$ matrices $U = \exp(i \vec \tau \!\cdot\! \vec \varphi\,) = \cos|\vec \varphi\,| + i\vec \tau \!\cdot\! \hat\varphi \, \sin|\vec \varphi\,|$\\ is extended to the $SU(3)$ group with eight real…
In this paper we introduce and study a family $\Phi_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k…
Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha…
For a function $f\colon \mathbb{N}\to\mathbb{N}$, define $N^{\times}_{f}(x)=\#\{n\leq x: n=kf(k) \mbox{ for some $k$} \}$. Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…
Given a finitely generated multiplicative subgroup of rational numbers $\Gamma$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group…
We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function,…
Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In…
We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log…
We obtain several asymptotic estimates for the sums of the restricted divisor function $$ \tau_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$ over short arithmetic progressions, which improve some results of J. Truelsen. Such…
Many exponential sums over finite fields, including Gauss sums and Kloosterman sums, arise as the Fourier transform with respect to a character of the trace function of an $\ell$-adic sheaf on a commutative algebraic group. We study the…
We use the resolution of singularities algorithm of [G4] to provide new estimates for exponential sums as well as new bounds on how often a function f(x) such as a polynomial with integer coefficients is divisible by various powers of a…
In this paper we establish three Barban-Davenport-Halberstam type theorems. Namely for exponential sums over primes, for Piatetski-Shapiro primes and for exponential sums over Piatetski-Shapiro primes.
We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…