The prime power conjecture is true for .
The probability that a complete intersection is smooth
Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection...
The probability that a random monic -adic polynomial splits.
The problema bovinum of Archimedes.
The problems of the non-Archimedean analysis generated by quantum physics
The product of linear forms in a field of series and the Riemann hypothesis for curves
The product of two Dirichlet series
The proof of Minkowski’s conjecture concerning the critical determinant of the region for p ≥ 6
The propinquity of divisors
The Pythagoras number of some affine algebras and local algebras.
The q-Mellin transform of automorphic forms and converse theorems
The quadratic form transfer and valuations.
The Quadratic Schur Subgroup over Local and Global Fields.
The Quality of the Diophantine Approximations Found by the Jacobi-Perron Algorithm and Related Algorithms.
The quantitative subspace theorem for number fields
The quickest proof of the prime number theorem
The R₂ measure for totally positive algebraic integers
Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates are positive real numbers. We study the set ₂ of the quantities . We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.
The radical factors of over finite fields.
The Ramsey-type version of a problem of Pomerance and Schinzel
The range of the sum-of-proper-divisors function
Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.