Document Type
Lecture
Publication Date
10-15-2018
Abstract
Let S(t) denote the argument of the Riemann zeta-function at the point (1/2) + it. Let Sₙ(t) be the n-th antiderivative of S(t) (adding a suitable constant cₙ at each step). In 1924, J. E. Littlewood established, under the Riemann hypothesis, that Sₙ(t) = O(log t / (log log t)^(n+1)), and this estimate has never been improved in its order of magnitude over the last 92 years. The efforts have focused on improving the implicit constant in this estimate. In this talk, we will show how to obtain the best (up to date) form of all of these estimates. This involves the use of certain special entire functions of exponential type. This is an application of approximation theory in analytic number theory. This is joint work with Emanuel Carneiro (IMPA).
Relational Format
presentation
Recommended Citation
Chavez, Andres Chirre, "Bounding Sₙ(t) on the Riemann Hypothesis" (2018). Algebra/Number Theory Seminar. 30.
https://egrove.olemiss.edu/math_algebra_number_theory/30
