10,629 research outputs found
The exceptional set for the number of primes in short intervals
We give upper bounds for the number of x up to X such that the interval (x, x+h) does not contain the expected quantity of primes. Here h is small with respect to x
Burgess's Bounds for Character Sums
We prove that Burgess's bound gives an estimate not just for a single
character sum, but for a mean value of many such sums.Comment: Minor changes and addition of reference to Gallagher & Montgomer
The distribution and moments of the error term in the Dirichlet divisor problem
This paper will consider results about the distribution and moments of some of the well known error terms in analytic number theory. To focus attention we begin by considering the error term ∆(x) in the Dirichlet divisor problem, which is defined a
Zeros of Systems of -adic Quadratic Forms
It is shown that a system of quadratic forms over a -adic
field has a non-trivial common zero as soon as the number of variables exceeds
, providing that the residue class field has cardinality at least .Comment: Revised version, with better treatment and results for characteristic
Subconvexity for a double Dirichlet series
For Dirichlet series roughly of the type
the subconvexity bound is proved on
the critical lines . The convexity bound would replace 1/6
with 1/4. In addition, a mean square bound is proved that is consistent with
the Lindel\"of hypothesis. An interesting specialization is in which
case the above result give a subconvex bound for a Dirichlet series without an
Euler product.Comment: 17 page
The largest prime factor of
The largest prime factor of has been investigated by Hooley, who gave a conditional proof that it is infinitely often at least as large as , with a certain positive constant . It is trivial to obtain such a result with . One may think of Hooley's result as an approximation to the conjecture that is infinitely often prime. The condition required by Hooley, his R conjecture, gives a non-trivial bound for short Ramanujan-Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of is infinitely often at least as large as , though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non-trivial bound for short Ramanujan-Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus
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