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The Bombieri–Vinogradov theorem for exponential sums over primes [PDF]

Notes on Number Theory and Discrete Mathematics
In this paper, we revisit Lemma 18 from [2], which concerns a Bombieri–Vinogradov type theorem for exponential sums over primes. We provide a corrected version of the lemma, clarify the original arguments, and address certain inaccuracies present in the ...
Stoyan Dimitrov
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Exponential sums over primes in short intervals

Journal of Number Theory, 2015
Let Λ(n) be the von Mangoldt function, x real and 2 ≤ y ≤ x.
Bingrong Huang
exaly   +4 more sources

Explicit relations between primes in short intervals and exponential sums over primes [PDF]

Functiones Et Approximatio, Commentarii Mathematici, 2014
one reference ...
Alessandro Zaccagnini, Alessandro Languasco   +1 more
exaly   +6 more sources

Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

Open Mathematics, 2017
In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., N=p13+…+pj3$\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with |pi−(N ...
Feng Zhao
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A Bombieri–Vinogradov-type result for exponential sums over Piatetski-Shapiro primes

Lithuanian Mathematical Journal, 2022
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}
exaly   +3 more sources

Brun's Fundamental Lemma and Exponential Sums over Primes

Journal of Number Theory, 2001
Using only a fundamental lemma in sieve methods together with Chebyshev's estimate for the average value for the von Mangoldt function \(\Lambda(n)\), the author establishes the bound for the exponential sum \[ \sum_{n\leq x}\Lambda(n)e^{2\pi i\alpha n}\ll x \sqrt{{d(q)\log^3q\over \varphi(q)}}, \] where \(d(q)\) is the divisor function and \(\varphi(q)
exaly   +3 more sources

EXPONENTIAL SUMS OVER PRIMES IN SHORT INTERVALS AND AN APPLICATION TO THE WARING–GOLDBACH PROBLEM [PDF]

Mathematika, 2016
Let $Λ(n)$ 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 \[ S_k(x,y;α) = \sum_{x< n \leq x+y} Λ(n) e\left( n^k α\right) \] when $k\geq 3$ for $α$ in the minor arcs.
Bingrong Huang
exaly   +3 more sources

Pair Correlation of Zeros, Primes in Short Intervals and Exponential Sums over Primes

Journal of Number Theory, 2000
Let \(S(\alpha) = \sum_{n\leq X}\Lambda(n)e(n\alpha)\), where \(e(x)=e^{2\pi ix}\), and \(\Lambda(n)\) is the von Mangoldt function defined to be \(\log p\) if \(n=p^m\), \(p\) a prime, \(m\geq 1\), and zero otherwise. This paper is concerned with the short \(L^2\) mean \[ S(X,\xi) = \int_{-\xi}^\xi |S(\alpha)|^2 d\alpha , \] with \(0\leq \xi \leq {1 ...
Alessandro Languasco
exaly   +4 more sources

On exponential sums involving Fourier coefficients of cusp forms over primes

Journal of Number Theory, 2016
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
exaly   +3 more sources

The Bombieri-Vinogradov theorem for nilsequences

Discrete Analysis, 2021
The Bombieri-Vinogradov theorem for nilsequences, Discrete Analysis 2021:21, 55 pp. The prime number theorem asserts that the density of the primes in the vicinity of a large integer $n$ is approximately $1/\log n$, or equivalently that the number of ...
Xuancheng Shao, Joni Teräväinen
doaj   +1 more source