Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(T) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(T) is shown to be rather regular and non-random; we give expressions for the average root number in this case.

Let E be a one-parameter family of elliptic curves over a number field K. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any non-degenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed explicitly in terms of E. The first such conjecture – commonly associated with Chowla – asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial. More precisely: given a homogeneous polynomial f ∈ Z[x, y], it is believed that µ(f(x, y)) averages to zero. This conjecture can be said to represent the parity problem in its pure form, while covering the same notional ground as the Bunyakovsky-Schinzel and Hardy-Littlewood conjectures taken together. For deg f = 1 and deg f = 2, Chowla’s conjecture is essentially equivalent to the prime number theorem. For deg f> 2, the conjecture has been unproven up to now; the traditional approaches by means of analysis and sieve theory fail. We prove the conjecture for deg f = 3. There remains to state the second arithmetical conjecture referred to previously. It is believed that any non-constant homogeneous polynomial f ∈ Z[x, y] yields to a square-free sieve. We sharpen the existing bounds on the known cases by a sieve refinement and a new approach combining height functions, sphere packings and sieve methods. iii ��������������������� � ��������� � ����� � �����

A square-free sieve is a result that gives an upper bound for how often a square-free polynomial may adopt values that are not square-free. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a square-free polynomial.

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After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian cubic fields. Then some conditional results proven elsewhere are discussed about totally real number fields of a fixed degree, each of whose normal closure has the symmetric group as Galois group. 1 Introduction. It was Littlewood who first addressed the question of how large the class number h of an imaginary quadratic field Q ( √ d) can be as a function of |d | as d → − ∞ through fundamental discriminants. In 1927 [14] he showed, assuming the generalized Riemann hypothesis (GRH), that for all fundamental d < 0

Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum

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Let f ∈ Z[x] be a polynomial of degree r ≥ 3 without roots of multiplicity r or (r − 1). Assume that f(x) ̸ ≡ 0 mod p r−1 has a solution in (Z/p r−1) ∗ for every p. Erdős conjectured that f(p) is then free of (r − 1)th powers for infinitely many primes p. This is proved here for every f a root of which generates its splitting field, and for some other f as well. The proof takes advantage of a certain kind of repulsion among integer points in curves of positive genus. Probabilistic arguments also come into play.