On the zeros of Epstein zeta-function, Kloosterman sums and some other problems of number theory

In the talk, we will give a brief survey of some recent results concerning the distribution of zeros of special $L$-functions. In particular, we present the theorem asserting that the positive proportion of zeros of Epstein zeta-function lies on the critical line. It means that the critical line is still a “special” set for this function despite the fact that it has infinitely many zeros lying outside this line. In the second part, we will speak about the estimates of incomplete Kloosterman sums and theirs generalizations. In particular, we will consider Kloosterman sums over prime numbers. Such sums can be estimated without using a classical bound of A. Weil.