Sums of One Prime Power and Four Prime Cubes in Short Intervals

Enumeration of Power Sums Modulo A Prime

We consider, for odd primes p, the function N(p, m, e) which equals the number of subsets SC{1 ,...,p 1) with the property that &sxm = OL (modp). We obtain a closed form expression for N(p, m, IX). We give simple explicit formulas for N(p, 2, CZ) (which in some casea involve class numbers and fundamental units), and show that for a fixed m, the diffenx~ce between N(p, m, a) and its average valu...

Reciprocal Power Sums Modulo a Prime

Using a sum-product result due to Bourgain, Katz, and Tao, we show that for every 0 < ǫ ≤ 1, and every power k ≥ 1, there exists an integer N = N(ǫ, k), such that for every prime p and every residue class a (mod p), there exist positive integers x1, ..., xN ≤ p ǫ satisfying a ≡ 1 x1 + · · · + 1 xN (mod p). This extends a result of I. Shparlinski [5].

Computing prime harmonic sums

We discuss a method for computing ∑ p≤x 1/p, using time about x2/3 and space about x1/3. It is based on the Meissel-Lehmer algorithm for computing the prime-counting function π(x), which was adapted and improved by Lagarias, Miller, and Odlyzko. We used this algorithm to determine the first point at which the prime harmonic sum first crosses 4.

Squarefree Integers without Large Prime Factors in Short Intervals

Prime Numbers in Logarithmic Intervals

Let X be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type (p, p+ h], where p ≤ X is a prime number and h = o(X). Then we will apply this to prove that for every λ > 1/2 there exists a positive proportion of primes p ≤ X such that the interval (p, p+λ logX] contains at least a prime number. As a consequence we improve Cheer an...