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Distribution of Lattice Points Visible from the Origin
Abstract:Let Ω be a region in the plane which contains the origin, is star-shaped with respect to the origin and has a piecewise C1 boundary. For each integer Q≥ 1, we consider the integer lattice
On the structure of the irreducible polynomials over local fields
The paper is concerned with the structure of irreducible polynomials in one variable over a local field (K, v). The main achievement is the definition of a system P(ƒ) of invariant factors for each
The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit
We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of
The distribution of spacings between the fractional parts of n2α
We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is "badly approximable" by rationals, then the sequence of fractional parts
The distribution of spacings between fractional parts of lacunary sequences
Given an increasing sequence of integers a(n), it is known (due to Weyl) that for almost all reals t, the fractional parts of the dilated sequence t*a(n) are uniformly distributed in the unit
On the index of Farey sequences
We prove some asymptotic formulae concerning the distri- bution of the index of Farey fractions of order Q as Q ! 1.
On the distribution of imaginary parts of zeros of the Riemann zeta function
Abstract We investigate the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta
Integrals of Eisenstein series and derivatives of L-functions
In his lost notebook, Ramanujan recorded a formula relating a “character analogue” of the Dedekind eta-function, the integral of a quotient of eta-functions, and the value of a Dirichlet Lfunction at
Explicit formulas for the pair correlation of zeros of functions in the Selberg class
For any two functions F and G in the Selberg class we prove explicit formulas which relate sums over pairs of zeros, of the form: P ÿTUgF ; gGUT f …rF ‡ rG† to sums over prime powers, of the form: T