Arithmetic of formal groups and applications I: Universal norm subgroups.
Arithmetic of half integral weight theta-series
Arithmetic of Hermitian forms.
Arithmetic of linear forms involving odd zeta values
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of and , as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers , , , and is irrational.
Arithmetic of non-principal orders in algebraic number fields
Let be an order in an algebraic number field. If is a principal order, then many explicit results on its arithmetic are available. Among others, is half-factorial if and only if the class group of has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.
Arithmetic of Pell surfaces
Arithmetic of quaternary quadratic forms
Arithmetic of the modular function
We find a generator of the function field on the modular curve X₁(4) by means of classical theta functions θ₂ and θ₃, and estimate the normalized generator which becomes the Thompson series of type 4C. With these modular functions we investigate some number theoretic properties.
Arithmetic on blocks.
Arithmetic on elliptic curves with complex multiplication. II.
Arithmetic over the rings of all algebraic integers.
Arithmetic progressions and the primes.
We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.
Arithmetic progressions formed by pseudoprimes
Arithmetic progressions in a unique factorization domain
Arithmetic progressions in sums of subsets of sparse sets
Arithmetic progressions in sumsets
1. Introduction. Let A,B ⊂ [1,N] be sets of integers, |A|=|B|=cN. Bourgain [2] proved that A+B always contains an arithmetic progression of length . Our aim is to show that this is not very far from the best possible. Theorem 1. Let ε be a positive number. For every prime p > p₀(ε) there is a symmetric set A of residues mod p such that |A| > (1/2-ε)p and A + A contains no arithmetic progression of length (1.1). A set of residues can be used to get a set of integers in an obvious way. Observe...
Arithmetic progressions of arbitrary length and consisting only of primes and almost primes.
Arithmetic progressions of length three in subsets of a random set
Arithmetic progressions of prime-almost-prime twins
Arithmetic progressions of primitive roots of a prime. II.