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We strengthen the recent result of Cherubini and Guerreiro on the square mean of the error term in the prime geodesic theorem for PSL 2 ( Z ) . We also develop a short interval version of this result.

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions., Comment: 8 pages

Let Λ ( n ) be the von Mangoldt function, x real and 2 ≤ y ≤ x . This paper improves the estimate on the exponential sum over primes in short intervals S k ( x , y ; α ) = ∑ x n ≤ x + y Λ ( n ) e ( n k α ) when k ≥ 4 for all α ∈ [ 0 , 1 ] . And then combined with the Hardy–Littlewood method, this enables us to give some short interval variants of Hua's theorems in additive prime number theory.

The main purpose of this paper is using the analytic methods to study the hybrid mean value involving the character sums, general quadratic Gauss sums and general Kloosterman sums, and give several interesting mean value formulae.

Let Ψ(x, y) denote the number of positive integers ≦ x and free of prime factors > y. De Bruijn showed that the relation Ψ(x, x 1 u ) ∼ xϱ(u) holds, as x → ∞, uniformly in the range 1 ≦ u ≦ ( log x) 3 8 − ϵ . We extend this range to 1 ≦ u ≦ log x ( log log x) 5 3 + ϵ , and give a similar, but weaker estimate for the range 1 ≦ u ≦ log x (1 + ϵ) log log x . We also prove a short interval result.

An elementary proof is given that there exists a constant c2 = c2(k) such that for x sufficiently large (depending on k), the interval (x, x + c2x0], where θ = 5 (10k + 1) , contains a k-free number. This result improves on a previous result of Halberstam and Roth [J. London Math. Soc. (2) 26 (1951), 268–273] .

Elementary and exponential sum approaches are used to obtain results for gaps between squarefree numbers. An elementary proof is given that there exists a positive constant c such that if x is sufficiently large, then the interval (x, x + h], where h = cx 2 9 , contains a squarefree number. Using exponential sums, it is shown that for any e > 0 one can take h = x 47 217 + e .