Results 11 to 20 of about 24,946 (242)
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
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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
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On the Distribution of Atkin and Elkies Primes [PDF]
, 2011 Given an elliptic curve E over a finite field F_q of q elements, we say that an odd prime ell not dividing q is an Elkies prime for E if t_E^2 - 4q is a square modulo ell, where t_E = q+1 - #E(F_q) and #E(F_q) is the number of F_q-rational points on E ...
Shparlinski, Igor E. +1 more
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Exponential Sums Over Primes in an Arithmetic Progression [PDF]
Proceedings of the American Mathematical Society, 1985 In 1979 A. F. Lavrik obtained some estimates for exponential sums over primes in arithmetic progressions by an analytic method. In the present paper we give an estimate for the same sums, comparable with Lavrik’s estimate, by means of elementary methods like Vaughan’s identity.
A. BALOG, PERELLI, ALBERTO
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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.
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Primes in short arithmetic progressions
, 2003 We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes.
Schlage-Puchta, Jan-Christoph
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New proof and generalization of some results on translated sums over k-almost primes
Comptes Rendus. MathématiqueA sequence $\mathcal{A}$ of strictly positive integers is said to be primitive if none of its terms divides the others, Erdős conjectured that the sum $f(\mathcal{A},0)\le f(\mathbb{N}_{1},0),$ where $\mathbb{N}_{1}$ is the sequence of prime numbers and $
Laib, Ilias
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Exterior powers in Iwasawa theory
, 2021 The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion.
Bleher, F. +5 more
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On Erdős sums of almost primes
Comptes Rendus. MathématiqueIn 1935, Erdős proved that the sums $f_k=\sum _n 1/(n\log n)$, over integers $n$ with exactly $k$ prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that $f_k$ is maximized by the prime sum $f_1=\sum _p 1/(p\log p)$.
Gorodetsky, Ofir +2 more
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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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