Results 21 to 30 of about 9,690 (225)
Restriction theory of the Selberg sieve, with applications [PDF]
, 2004 The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L^2-L^p restriction theorem for majorants of this type.
Green, Ben, Tao, Terence
core +4 more sources
On certain exponential sums over primes
Journal of Number Theory, 2009 Let \(f(x)\) be a real valued polynomial of degree \(k\geq 4\) and irrational leading coefficient \(\alpha\). Exponential sums of the form \[ S:=\sum_{p\leq N} (\log p) e(f(p)) \] have received a lot of interest. \textit{G. Harman} proved in [Mathematika 28, 249--254 (1981; Zbl 0465.10029)] that if \(q\) is the denominator of a convergent of \(\alpha\),
Maier, H., Sankaranarayanan, A.
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Fractional parts of polynomials over the primes [PDF]
, 2017 Let f be a polynomial with irrational leading coefficient. We obtain inequalities for the distance from the nearest integer of f(p) that hold for infinitely many primes p.
Baker, Roger
core +1 more source
Explicit upper bounds for exponential sums over primes [PDF]
Mathematics of Computation, 2000 We give explicit upper bounds for linear trigonometric sums over primes.
Daboussi, Hedi, Rivat, Joel
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Double Character Sums over Subgroups and Intervals [PDF]
, 2014 We estimate double sums $$ S_\chi(a, I, G) = \sum_{x \in I} \sum_{\lambda \in G} \chi(x + a\lambda), \qquad 1\le a < p-1, $$ with a multiplicative character $\chi$ modulo $p$ where $I= \{1,\ldots, H\}$ and $G$ is a subgroup of order $T$ of the ...
Bourgain +7 more
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On Sums of Powers of Almost Equal Primes [PDF]
, 2014 We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$.
Wei, Bin, Wooley, Trevor D.
core +5 more sources
Pseudorandomness and Dynamics of Fermat Quotients [PDF]
, 2010 We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$.
Ostafe, Alina, Shparlinski, Igor E.
core +1 more source
Real exponential sums over primes and prime gaps
, 2023 We prove that given $λ\in \R$ such that $0 < λ< 1$, then $π(x + x^λ) - π(x) \sim \displaystyle \frac{x^λ}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short intervals. In particular, we give a positive answer (for all sufficiently large number) to some old conjectures about prime numbers, such as Legendre's
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Mean values of Dirichlet polynomials and applications to linear equations with prime variables
, 2004 We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes.
Angel V. Kumchev +2 more
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EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES [PDF]
Mathematika, 2011 Summary: We consider exponential sums with x-coordinates of points \(qG\) and \(q^{-1}G\) where \(G\) is a point of order \(T\) on an elliptic curve modulo a prime \(p\) and \(q\) runs through all primes up to \(N\) (with \(\gcd(q,T)=1\) in the case of the points \(q^{-1}G\)).
Ostafe, Alina, Shparlinski, Igor E
openaire +3 more sources