Title:

Some problems involving prime numbers

The first problem we consider is a variation of the PiatetskiShapiro Prime Number Theorem. Consider a function g(y), growing faster than linearly. We ask how often is the integer part of a function g(y) no less than some distance j from a prime number Using Huxley's method of exponential sums the investigation shows how the rate at which g(y) increases is dependent on the size of j. The faster g(y) increases, the larger the value of j. The second problem investigates primes of arithmetic progressions, a mod g, in short intervals of the form (x, x+xe), where x is sufficiently large in terms of q, cp < x for some 77 > 0. Such a result was proved by Fogels, for some 6 < 1. We explicitly determine the relationship between 6 and 77 to establish admissible values for both. Lastly we use our version of Fogels' theorem and a variation of Vaughan's treatment of the ctp problem to investigate the following problem. Given a real number a in the interval (0,1) how many Farey fractions of the Farey sequence of order Q do we have to pass to go from a to a Farey fraction with prime denominator
