506 research outputs found
Notes on Pair Correlation of Zeros and Prime Numbers
These notes are based on my four lectures given at the Newton Institute in
April 2004 during the Recent Perspectives in Random Matrix Theory and Number
Theory Workshop. Their purpose is to introduce the reader to the analytic
number theory necessary to understand Montgomery's work on the pair correlation
of the zeros of the Riemann zeta-function and subsequent work on how this
relates to prime numbers. A very brief introduction to Selberg's work on the
moments of is also given
Are There Infinitely Many Primes?
This paper is based on a talk given to motivated high school (and younger)
students at a BAMA (Bay Area Math Adventure) event. Some of the methods used to
study primes and twin primes are introduced.Comment: 21 page
On the norm of an exponential sum involving the divisor function
In this paper, we obtain bounds on the norm of the sum where is the divisor function
A note on S(T) and the zeros of the Riemann zeta-function
Let denote the argument of the Riemann zeta-function at the point
. Assuming the Riemann Hypothesis, we sharpen the constant in the
best currently known bounds for and for the change of in
intervals. We then deduce estimates for the largest multiplicity of a zero of
the zeta-function and for the largest gap between the zeros.Comment: 5 page
Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem
We calculate the triple correlations for the truncated divisor sum
. The 's behave over certain averages just as
the prime counting von Mangoldt function does or is conjectured to
do. We also calculate the mixed (with a factor of ) correlations.
The results for the moments up to the third degree, and therefore the
implications for the distribution of primes in short intervals, are the same as
those we obtained (in the first paper with this title) by using the simpler
approximation . However, when is used the
error in the singular series approximation is often much smaller than what
allows. Assuming the Generalized Riemann Hypothesis for
Dirichlet -functions, we obtain an -result for the variation
of the error term in the prime number theorem. Formerly, our knowledge under
GRH was restricted to -results for the absolute value of this
variation. An important ingredient in the last part of this work is a recent
result due to Montgomery and Soundararajan which makes it possible for us to
dispense with a large error term in the evaluation of a certain singular series
average. We believe that our results on 's and
's can be employed in diverse problems concerning primes.Comment: 51 page
On the differences between consecutive prime numbers, I
We show by an inclusion-exclusion argument that the prime -tuple
conjecture of Hardy and Littlewood provides an asymptotic formula for the
number of consecutive prime numbers which are a specified distance apart. This
refines one aspect of a theorem of Gallagher that the prime -tuple
conjecture implies that the prime numbers are distributed in a Poisson
distribution around their average spacing.Comment: 6 page
The Path to Recent Progress on Small Gaps Between Primes
This paper describes some of the ideas used in the development of our work on
small gaps between primes.Comment: 11 page
Primes in Tuples II
We prove that there are infinitely often pairs of primes much closer than the
average spacing between primes - almost within the square root of the average
spacing. We actually prove a more general result concerning the set of values
taken on by the differences between primes.Comment: 35 page
Primes in Tuples IV: Density of small gaps between consecutive primes
We show that a positive proportion of all gaps between consecutive primes are
small gaps. We provide several quantitative results, some unconditional and
some conditional, in this flavour
Primes in Tuples I
We introduce a method for showing that there exist prime numbers which are
very close together. The method depends on the level of distribution of primes
in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we
prove that there are infinitely often primes differing by 16 or less. Even a
much weaker conjecture implies that there are infinitely often primes a bounded
distance apart. Unconditionally, we prove that there exist consecutive primes
which are closer than any arbitrarily small multiple of the average spacing,
that is, This last
result will be considerably improved in a later paper.Comment: 36 page
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