Andrew Granville's Home Page

2003 Publications

The number of fields generated by the square root of values of a given polynomial (with Pam Cutter and Tom Tucker)
Canadian Mathematical Bulletin, 46 (2003), 71-79.

The \( abc\)-conjecture is applied to various questions involving the number of distinct fields \( \mathbb Q(\sqrt{f(n)})\), as one varies over integers \( n\).


Decay of mean-values of multiplicative functions (with K. Soundararajan)
Canadian Journal of Mathematics, 55 (2003), 1191-1230.

We give an explicit version of Halasz's Theorem, giving an upper bound on the mean value of multiplicative functions in terms of its "distance" from any function of the form \( n^{it}\). We give examples to show our bounds are best possible within a factor of 10.


Distribution of values of \( L(1,\chi_d) \) (with K. Soundararajan)
Geometric and Functional Analysis, 13 (2003), 992-1028; (Errata) 14 (2003), 245-246.

We prove various results (including some conjectures of Montgomery and Vaughan) on the distribution of the values of \( L(1,\chi_d)\), including finding complex moments and extreme values.