Related papers: On Weyl sums over primes in short intervals

In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.

In this paper we study Weyl sums over friable integers (more precisely $y$-friable integers up to $x$ when $y = (\log x)^C$ for a large constant $C$). In particular, we obtain an asymptotic formula for such Weyl sums in major arcs,…

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…

Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem and Mertens' theorems using a smoothing argument. Our results improve upon previous bounds in both…

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we…

In this paper we first establish new explicit estimates for Chebyshev's $\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime…

We present a hybrid approach to bounding exponential sums over kth powers via Vinogradov's mean value theorem, and derive estimates of utility for exponents k of intermediate size.

In this paper, we study the $L^p$ maximal estimates for the Weyl sums $\sum_{n=1}^{N}e^{2\pi i(nx + n^{k}t)}$ with higher-order $k\ge3$ on $\mathbb{T}$, and obtain the positive and negative results. Especially for the case $k=3$, our result…

We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other…

In this paper, the formulas of some exponential sums over finite field, related to the Coulter's polynomial, are settled based on the Coulter's theorems on Weil sums, which may have potential application in the construction of linear codes…

We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if $t$ is an integer different from $0, 1$ or -1 and if $\A \subset \Zp$ is not too large (with respect to $p$),…

In this paper we generalize a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefcients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial…

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

In the paper, the authors establish some interesting identities and inequalities involving the extended Weyl type fractional integrals.

We obtain an estimate for the cubic Weyl sum which improves the bound obtained from Weyl differencing for short ranges of summation. In particular, we show that for any $\varepsilon>0$ there exists some $\delta>0$ such that for any coprime…

This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently…

We prove two estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker.

The author sharpens a result of Jia (1996), showing that the interval $[n, n+n^{\frac{1}{21.5}+\varepsilon}]$ contains prime numbers for almost all $n$. Watt's mean value bound, a delicate sieve decomposition and more accurate estimates for…