Non-trivial estimates for upper bounds of the absolute values of incomplete Kloosterman sums of the form $\sum_{n\le x,(n,m)=1}\exp(2\pi i\frac{an^*+bn}{m})$, $(a,m)=1$, $\exp((\log m)^{4/5+\epsilon})\le x \le m^{4/7}$; $\sum_{x<n\le x+h,(n,m)=1}\exp(2\pi i\frac{a_1 n^*+b_1 n}{m})$, $(a_1,m)=1$, $\exp((\log m)^{5/6+\epsilon})\le x \le m^{4/7}$, $x\exp{(-\frac{(\log x)^4}{(\log m)^3 (\log\log m)^{44}})}\le h \le 0.5x$ are established. A. Ghosh's result for the number of sign-changes of the function $S(t)$, $S(t)=\frac{1}{\pi}\arg\zeta(1/2+it)$, on segments of the form $T\le t\le T+ H$, $H=T^a$, $a>27/82$.

Graduated from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 2000 (department of mathematical analysis).

1. Korolev M. A., “Nepolnye summy Kloostermana i ikh prilozheniya”, Izv. RAN. Ser. matem., 64:6 (2000), 41–64      
2. Korolev M. A., “O raspredelenii obratnykh velichin po zadannomu modulyu”, Sovremennye issledovaniya v matematike i mekhanike, Trudy XXIII Konferentsii molodykh uchenykh mekhaniko-matematicheskogo fakulteta MGU, T. II (9–14 aprelya 2001 g.), Izd-vo Tsentra prikladnykh issledovanii pri mekhaniko-matematicheskom fakultete MGU, M., 2001, 184–186

• Steklov Mathematical Institute, Russian Academy of Sciences
• M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics