Let c q (n) denote the Ramanujan sum modulo q, and let x and y be large reals, with x = o(y). We obtain asymptotic formulas for the sums n≤y q≤x c q (n) k (k = 1, 2).
We consider the question of approximating any real number α by sums of n rational numbers a 1 q 1 + a 2 q 2 + · · · + an qn with denominators 1 q 1 , q 2 , . . . , q n N. This leads to inquiries on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator as well as by rational numbers with smooth denominator.
We study short intervals which contain an "almost square", an integer n that can be factored as n = ab with a, b close to √ n. This is related to the problem on distribution of n 2 α (mod 1) and the problem on gaps between sums of two squares.
Goldston and Montgomery [3] proved that the Strong Pair Correlation Conjecture and two second moments of primes in short intervals are equivalent to each other under Riemann Hypothesis. In this paper, we get the second main terms for each of the above and show that they are almost equivalent to each other.
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