On the composition of nondegenerate quadratic forms with an arbitrary index
On the composition of some arithmetic functions. II.
On the composition of the arithmetic functions σ and φ
On the composition of the Euler function and the sum of divisors function
Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.
On the compositum of all degree extensions of a number field
We study the compositum of all degree extensions of a number field in a fixed algebraic closure. We show contains all subextensions of degree less than if and only if . We prove that for there is no bound on the degree of elements required to generate finite subextensions of . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of , but that one can take when is prime. This question was inspired by work of Bombieri and...
On the computation of Hermite-Humbert constants for real quadratic number fields
We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields and . Finally we compute the Hermite-Humbert constant for the number field .
On the computation of quadratic -class groups
We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer mod and the factorization of , computes the structure of the -Sylow subgroup of the class group of the quadratic order of discriminant in random polynomial time in .
On the computation of the class numbers of some cubic fields.
On the concentration of certain additive functions
We study the concentration of the distribution of an additive function f when the sequence of prime values of f decays fast and has good spacing properties. In particular, we prove a conjecture by Erdős and Kátai on the concentration of when c > 1.
On the concentration of points on modular hyperbolas and exponential curves
On the concept of optimality interval.
On the conductor discriminant formula.
On the conductor formula of Bloch
In [6], S. Bloch conjectures a formula for the Artin conductor of the ℓ-adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The formula, which we call the conductor formula of Bloch, enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms. In this paper, we prove the formula in arbitrary dimension under the assumption that the reduced closed fiber has normal crossings.
On the conductor of the Jacobi sum Hecke character
On the congruence
On the congruence modulo .
On the congruence , where q is a prime and f is a polynomial
On the congruence , where q is a prime and f is a k-normal polynomial
On the congruence f(x) + g(y) + c ≡ 0 (mod xy) (completion of Mordell's proof)
Assertions on the congruence f(x) + g(y) + c ≡ 0 (mod xy) made without proof by Mordell in his paper in Acta Math. 88 (1952) are either proved or disproved.
On the congruence .