On the variance of squarefree integers in short intervals and arithmetic progressions

Arithmetic Progressions of Primes in Short Intervals

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: If N is sufficiently large and M is not too small compared with N , then the primes in the interval [N, N + M ] contains many arithmetic progressions of length k.

Squarefree polynomials and Möbius valuesin short intervalsand arithmetic progressions

Abstract. We calculate the mean and variance of sums of the Möbius function μ and the indicator function of the squarefrees μ, in both short intervals and arithmetic progressions, in the context of the ring Fq[t] of polynomials over a finite field Fq of q elements, in the limit q → ∞. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent eq...

Squarefree Integers without Large Prime Factors in Short Intervals

Integers without large prime factors in short intervals and arithmetic progressions

On Arithmetic Progressions in Sums of Sets of Integers

3 Proof of Theorem 1 9 3.1 Estimation of the g1 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Estimation of the g3 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Estimation of the g2 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Putting everything together. . . . . . . . . . . . . . . . . . . . . ....