Algebraic Values of Certain Analytic Functions

作者：Boxall Gareth J^*; Jones Gareth O

来源：International Mathematics Research Notices, 2015, (4): 1141-1158.

DOI：10.1093/imrn/rnt239

Building on recent work of Masser concerning algebraic values of the Riemann zeta function, we prove two general results about the scarcity of algebraic points on the graphs of certain restrictions of certain analytic functions. For any of the graphs to which our results apply and any positive integer d, we show that there are at most C(log H)(3+epsilon) algebraic points of degree at most d and multiplicative height at most H on that graph. In particular, we obtain this conclusion for any restriction of Gamma(z) or zeta(z)/pi(z) to a compact disk, answering questions from Masser's paper, the latter having been suggested by Pila. As in Masser's original work, the constant C may be effectively computed from certain data associated with the function in question.

• 出版日期2015