Primes and divisors

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AbstractTitle: Primes and divisorsThis proposal is concerned with three projects related to the distributionof prime numbers and divisors of integers. The first project concernscontinuing investigations (jointly with S. Konyagin) of subtleinequities in the distribution of primes in arithmetic progressions.Such problems are intimately connected with the distribution of non-trivialzeros of Dirichlet L-functions and the Riemann zeta function. The researchfocus is on further illuminating the influence of hypothetical zeros of theL-functions having real part different from 1/2.A second project deals with the limitations of sieve methods, which arepowerful techniques for bounding the number of primes (or numbers witha bounded number of prime factors) in an integer sequence.The goal of the proposer's work is an improved understanding ofhow the distribution of primes in a sequence is influenced by thedistribution of the sequence in arithmetic progressions and the distributionof the sequence on numbers with an odd number (respectively and even number)of prime factors.The final project is to improve bounds for the density of integers possessinga divisor in a given interval, a problem which is central to the theory ofthe distribution of divisors of integers.The study of prime numbers is more than 2000 years old, and the most importantproblems revolve around how the primes are distributed. Besides applicationsto many other branches of mathematics, prime number theory is crucial tomodern coding theory and cryptography, e.g. secure Internet communication.The proposed research deals with questions concerning how the primes aredistributed within certain sets of integers, and questions on the relatedsubject of the statistical behavior of the distribution of divisors ofnumbers. The proposer will involve graduate students,post-docs and/or talented undergraduate students (REU) in these researchprojects.