| Title |
Joint discrete universality in the selberg-steuding class and non-trivial zeroes of the riemann zeta-function / |
| Translation of Title |
Selbergo-Štoidingo klasės diskretus jungtinis universalumas ir netrivialūs Rymano dzeta funkcijos nuliai. |
| Authors |
Togobickij, Benjaminas |
| Full Text |
|
| Pages |
35 |
| Keywords [eng] |
approximation, discreteness, joint universality, non-trivial zeroes, Riemann zeta-function, Selberg-Steuding class, weak convergence, universality. |
| Abstract [eng] |
In this thesis, the approximation property of a certain class of zeta-functions is studied. It is shown that $L$-functions from the Selberg-Steuding class $\mathscr{S}$ are jointly universal in the Voronin sense concerning discrete shifts involving the non-trivial zeroes of the Riemann zeta-function $\zeta(s)$, or, more precisely, we approximate simultaneously any collection of non-vanishing analytic functions on compact subsets by using shifts $L(s + i\gamma_kh_j)$ with accuracy $\eps>0$. Here $\gamma_k$ are the imaginary parts of the non-trivial zeroes of the function $\zeta(s)$. Also, a modification of this theorem is obtained, i.e., we extend the result to positive density and show that the limit exists for all but at most countably many $\eps>0$. These theorems under the weak Montgomery pair correlation conjecture and certain linear independence for the fixed $h_j$'s are shown. The proof involves the application of Mergelyan's approximation theorem and a limit theorem in the space of analytic functions. |
| Dissertation Institution |
Vilniaus universitetas. |
| Type |
Master thesis |
| Language |
English |
| Publication date |
2024 |
