Results 11 to 20 of about 9,640 (238)
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
The ternary Goldbach problem [PDF]
, 2014 The ternary Goldbach conjecture, or three-primes problem, states that every odd number $n$ greater than $5$ can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in the twentieth
Helfgott, Harald Andrés
core +3 more sources
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.
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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
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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
New estimates for exponential sums over multiplicative subgroups and intervals in prime fields [PDF]
Journal of Number Theory, 2020 Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 i z/p)$, which improves a result of Bourgain and Garaev (2009).
Daniel Di Benedetto +5 more
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Explicit relations between primes in short intervals and exponential sums over primes [PDF]
Functiones et Approximatio Commentarii Mathematici, 2014 one reference ...
LANGUASCO, ALESSANDRO, A. Zaccagnini
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Multiple Exponential and Character Sums with Monomials
, 2013 We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known ...
Shparlinski, Igor
core +1 more source