1,721,046 research outputs found
Applications of some exponential sums on prime powers: a survey.
No full textA survey paper on some recent results on additive problems with prime powers
The exceptional set in short intervals for two additive problems with primes: a survey
No full textWe give a brief account
about the exceptional sets in short intervals for the Goldbach and the
Hardy-Littlewood problems. In particular, we present two recent results about
Montgomery-Vaughan's type estimates for such exceptional sets
A conditional result of the exceptional set for Hardy-Littlewood numbers in short intervals
No full textAssuming the Generalized Riemann Hypothesis holds, we prove some conditional estimates on the exceptional set in short intervals for the Hardy-Littlewood problem
A note on primes and Goldbach numbers in short intervals
No full textLet be the Selberg
integral and the error term in Kaczorowski-Perelli's weighted form
of the classical explicit formula. We prove that the estimate
is connected with an appropriate estimate of
, uniformly for and in some
ranges. Moreover, assuming a suitable bound for the quantity
, we also obtain, for all sufficiently
large and , that every interval contains
Goldbach numbers
On the sum of a prime and a k-free number
No full textWe prove an asymptotic
formula (that refines old results by Walfisz and Mirsky) for the number of
representations of sufficiently large integer as a sum of a prime and a
-free number,
Some results on Goldbach's problem
Full text linkIn Section 1 we introduce the Goldbach Conjecture and give a brief account on the main contribution to this subject. In the other sections we sketch the proofs of some results on the existence of Goldbach numbers in short intervals and on the exceptional set for Goldbach's problem
On the exceptional set of Goldbach numbers in short intervals
No full textLet be the set of
integers which are not a sum of two primes, and
, where . A well known result of
Montgomery-Vaughan proves that there exists an absolute positive constant
such that . Here we prove that
there exists an absolute positive constant such that, for , , improving a
result by Peneva
Some refinements of error terms estimates for certain additive problems with primes
No full textWe study, under the
assumption of the Generalized Riemann Hypothesis, the individual and
mean-square error terms for the number of integers representable as a sum of
primes. We improve, using a smoothing technique,
Friedlander-Goldston's recent results on this topic. Moreover, we remark that
the argument we use can also be applied to other similar problems
On the exceptional set of Hardy-Littlewood's numbers in short intervals
Full text linkIn 1923 Hardy and Littlewood
conjectured that every sufficiently large integer is either a -power of an
integer or a sum of a prime and a -power of an integer, for . We
will call HL-numbers the integers that are a sum of a prime and a -power
of an integer. Let now and denote by the set of integers
which are neither an HL-number nor a power of an integer. Here we prove that
there exists an absolute positive constant such that for [ ert E_k(X,H)ert ll H^{1-delta/(5K)},
] where , thus improving previous results by Perelli-Pintz,
Mikawa, Perelli-Zaccagnini and Zaccagnini
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