Glasnik Matematicki, Vol. 48, No. 1 (2013), 59-65.

ON ZEROS OF SOME ANALYTIC FUNCTIONS RELATED TO THE RIEMANN ZETA-FUNCTION

Antanas Laurinčikas

Abstract.   For some classes of functions F, we obtain that the function F(ζ(s)), where ζ(s) denotes the Riemann zeta-function, has infinitely many zeros in the strip 1/2 < Re s < 1. For example, this is true for the functions sin ζ (s) and cos ζ (s).

2010 Mathematics Subject Classification.   11M41.

Key words and phrases.   Riemann zeta-function, universality, zero-distribution.

DOI: 10.3336/gm.48.1.05

References:

1. H. M. Bui, B. Conrey and M. P. Young, More than 41% of the zeros of the zeta function are on the critical line, Acta Arith. 150 (2011), 35-64.
   MathSciNet     CrossRef

2. X. Gourdon, The 10¹³ first zeros of the Riemann zeta function and zeros computation at very large height, http://numbers.computation.free.fr, 2004.

3. A. Ivić, The Riemann zeta-function, Wiley, New York, 1985.
   MathSciNet    

4. A. Laurinčikas, Limit theorems for the Riemann zeta-function, Kluwer, Dordrecht, 1996.
   MathSciNet     CrossRef

5. A. Laurinčikas, Universality of the Riemann zeta-function, J. Number Theory 130 (2010), 2323-2331.
   MathSciNet     CrossRef

6. A. Laurinčikas and R. Garunkštis, The Lerch Zeta-Function, Kluwer, Dordrecht, 2002.
   MathSciNet    

7. S. N. Mergelyan, Uniform approximations to functions of complex variable, Usp. Matem. Nauk 7 (1952), 31-122 (Russian).
   MathSciNet    

8. S. M. Voronin, Theorem of the "universality" of the Riemann zeta-function, Izv. Akad. Nauk SSSR. Ser. Mat. 39 (1975), 475-486 (Russian).

9. S. M. Voronin, Selected works: mathematics, (ed. A. A. Karatsuba), Publishing House MGTU Im. N. E. Baumana, Moscow, 2006 (Russian).

10. J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Vol. XX, Amer. Math. Soc. Coll. Publ. 1960.
    MathSciNet