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Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x) :::: (c + o(I)) logx loglogx log log log 10gx(loglog logx)-2, where c = e Y and y is Euler's constant. Here, this result is shown with c = coe Y where Co = 1.31256... is the solution of the equation 4/… (More)

- Adam J. Harper, Ashkan Nikeghbali, Helmut Maier, Maksym Radziwiłł
- 2015

- Paul Pollack, Helmut Maier
- 2014

Let σ denote the usual sum-of-divisors function. We show that every positive real number can be approximated arbitrarily closely by a fraction m/n with σ (m) = σ (n). This answers in the affirmative a question of Erd˝ os. We also show that for almost all of the elements v of σ (N), the members of the fiber σ −1 (v) all share the same largest prime factor.… (More)

- HELMUT MAIER
- 2014

Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure µ on R, with respect to which certain normalized cotangent sums are equidistributed.… (More)

Let dn = Pn+i ~Pn denote the nth gap in the sequence of primes. We show that for every fixed integer A; and sufficiently large T the set of limit points of the sequence {(dn/logra, ■ • • ,dn+k-i/logn)} in the cube [0, T]k has Lebesgue measure > c(k)Tk, where c(k) is a positive constant depending only on k. This generalizes a result of Ricci and answers a… (More)