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1. Baier, Stephan

et al.

Zhao, Liangyi

ON PRIMES IN QUADRATIC PROGRESSIONS2009In: International Journal of Number Theory, ISSN 1793-0421, Vol. 5, no 6, p. 1017-1035Article in journal (Refereed)

Abstract [en]

We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper by the authors [3].

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Eventually stable rational functions2017In: International Journal of Number Theory, ISSN 1793-0421, Vol. 13, no 9, p. 2299-2318Article in journal (Refereed)

Abstract [en]

For a field K, rational function phi is an element of K(z) of degree at least two, and alpha is an element of P-1(K), we study the polynomials in K[z] whose roots are given by the solutions in K to phi(n)(z) = a, where fn denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (phi, alpha) is called eventually stable over K. We conjecture that (phi, alpha) is eventually stable over K when K is any global field and a is any point not periodic under f (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when K has a discrete valuation for which (1) f has good reduction and (2) facts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of S-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.

If f is a non-constant polynomial with integer coefficients and q is an integer, we may regard f as a map from Z/qZ to Z/qZ. We show that the distribution of the (normalized) spacings between consecutive elements in the image of these maps becomes Poissonian as q tends to infinity along any sequence of square free integers such that the mean spacing modulo q tends to infinity.

In this paper, a special class of local zeta-functions is studied. The main theorem states that the functions have all zeros on the line R(s) = 1/2. This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have R(s) = 1/2.