Values of binary linear forms at prime arguments

value of a given binary linear form at prime arguments. Let λ₁ and λ₂ be positive real numbers such that λ₁/λ₂ is irrational and algebraic. For any (C, c) well-spaced sequence V and δ>0, let E(V, X, δ) denote the number of υ∈V with υ≤X for which the inequality |λ1p1+λ2ρ2−υ|<υ−δ has no solution in primes p₁, p₂. It is shown that for any ε>0,we have E(V, X, δ) «max(X35+2δ+ε,X23+43δ+ε).     

A remark on linear permutation polynomials

We give an explicit representation of the class of linear permutation polynomials. By the representation, the number of them can be computed easily.     

A remark on very ample linear series

We give a criterium on the existence of (e - 1)-very ample linear series on a general k-gonal curve of genus $g (e \geq 1)$, and we add some general remarks on such series.     

A remark on non-integral p-adic slopes for modular forms

We give a sufficient condition, namely “Buzzard irregularity”, for there to exist a cuspidal eigenform which does not have integral p-adic slope.     

Selberg''s integral and linear forms in zeta values

Using Selberg's integral, we present some new Euler-type integral representations of certain nearly poised hypergeometric series. These integrals are also shown to produce linear forms in odd and/or even zeta values that generalize previous work of the author.     

A lower bound for linear forms in values of a continued fraction

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β ₁ y ₁ + β ₂ y ₂ + α| is given, where y ₁ = y(x ₁), y ₂ = y(x ₂), x ₁ and x ₂ are positive integers, α, β ₁ and β ₂ are algebraic irrational numbers.     

A lower bound for linear forms in values of certain hypergeometric functions

Lower bounds are found for linear forms in values of certain hypergeometric functions and their derivatives, analogous to the bounds obtained by K. Mahler for the linear form of the numbers 1, e, e²,..., e^m.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 19–28, July, 1970.We wish to thank A. B. Shidlovskii for his interest in this work and for his many useful suggestions.     

A remark on the distribution of the values of the Riemann zeta function

Mathematical Notes - On a certain probability space, an analytic random element and a random variable both related to the Riemann zeta function and a measurable measure preserving transformation...     

Lower bounds for linear forms in values of polylogarithms

In this paper, we use Hermite-Padé approximations of the second kind to obtain a lower bound for the absolute value of a linear form, with integer coefficients, in values of polylogarithmic functions at a rational point. This estimate takes into account the growth of all coefficients of the linear form.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 623–629.Original Russian Text Copyright © 2005 by T. Hessami Pilehrood, H. Hessami Pilehrood.This revised version was published online in April 2005 with a corrected issue number.