Speaker: Yuta Suzuki

Venue: 203 Wentian Building, Weihai Campus

Time: May 8th, 16:00-16:50

Title: On error term estimate \`a la Walfisz for mean values of arithmetic functions

Abstract: A. Walfisz developed a method for error term estimate for mean values of arithmetic functions similar to the divisor summatory function, the Euler totient function, their powers, etc., and the general version of Walfisz's method was given by Balakrishnan and P\'etermann (1996). Walfisz's error term estimate for the Euler totient function has been the best possible for more than 50 years. However, recently, H.-Q. Liu (2016) improved Walfisz's estimate up to $(\log\log x)$-factor. H.-Q. Liu's main ingredient is to use Vaughan's identity instead of Vinogradov's original combinatorial decomposition.

 The aim of this talk is to extend the result of H.-Q. Liu to the general setting of Balakrishnan and P\'etermann. It turns out that the method of H.-Q. Liu cannot be used for the general case, so we develop some refined version of the Vinogaradov-style decomposition. We also succeeded in relaxing some conditions assumed by Balakrishnan and P\'etermann.