منابع مشابه
Primes in Almost All Short Intervals
It is well known that Huxley’s density estimates [5] for the zeros of the Riemann zeta-function yield J(x, h) = o(xh2(log x)−2), but only for h ≥ x1/6(log x) , for some C > 0. The weaker result with h ≥ x1/6+ε is proved in Saffari and Vaughan [8], Lemma 5, and in [13], where an identity of Heath-Brown (Lemma 1 of [3]) is used. This paper is inspired by Heath-Brown’s extension [4] of Huxley’s Th...
متن کاملOn the symmetry of primes in almost all short intervals
– In this paper we study the symmetry of primes in almost all short intervals; by elementary methods (based on the Large Sieve) we give, for h x log x (c > 0, suitable), a non-trivial estimate for the mean-square (over N < x ≤ 2N) of an average of “symmetry sums”; these sums control the symmetry of the von-Mangoldt function in short intervals around x. We explicitly remark that our results are ...
متن کاملPrimes in Short Intervals
Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x + H) − ψ(x), for 0 ≤ x ≤ N , is approximately normal with mean ∼ H and variance ∼ H logN/H , when N ≤ H ≤ N1−δ .
متن کاملPrimes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function
Abstract We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in “almost all” short intervals. It is well known that a relation like ψ(x)−ψ(x−y) ∼ y holds for almost all x ∈ [N, 2N ] in a range for y that depends on the width of the available zero-free regions for the Riemann zeta-function, and also on the strength of density b...
متن کاملPrimes in Beatty Sequences in Short Intervals
In this paper we show that sieve methods used previously to investigate primes in short intervals and corresponding Goldbach type problems can be modified to obtain results on primes in Beatty sequences in short intervals.
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1998
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-84-3-225-244