| Abstract [eng] |
In this dissertation the Lerch zeta-function, its derivative and the periodic Hurwitz zeta-function are studied. These functions are generalizations of the famous Riemann zeta-function. To get better understanding of the zero and a-value distribution of the periodic Hurwitz zeta-function we find the asymptotic formula for the number of nontrivial zeros and a-values. Prove, that the zeros and a-values of the periodic Hurwitz zeta-function are mostly clustered around the critical line. We conclude, that the number of zeros and a-values of the periodic Hurwitz zeta-function till the given size mainly depend on T, parameters and properties of the periodical sequence. For the derivative of the Lerch zeta-function we indicate zero-free regions; locate approximate positions of the trivial zeros; consider the asymptotic formula for the number of nontrivial zeros; explore the zero distribution with respect to the critical line. Calculations show that, for special case of the Lerch zeta-function when parameters are equal, the nontrivial zeros either lie extremely close to the critical line or are distributed almost symmetrically with respect to it. We investigate this phenomenon theoretically proving, that on average they are symmetrically distributed with a small error term. For this special case, there is the Speiser type relation between zeros of the Lerch zeta-function and its derivative. |
