Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Zeros of $ \zeta \sp{'} (s)$ in the critical strip


Author: Robert Spira
Journal: Proc. Amer. Math. Soc. 35 (1972), 59-60
MSC: Primary 10H05
DOI: https://doi.org/10.1090/S0002-9939-1972-0296035-5
MathSciNet review: 0296035
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the abscissa of convergence for the Dirichlet series $ {( - 1)^k}{(1 - {2^{1 - s}})^{k + 1}}{\zeta ^{(k)}}(s)$ is zero, where $ \zeta (s)$ is the Riemann zeta function. This implies the existence of infinitely many zeros of $ \zeta '(s)$ in the critical strip.


References [Enhancements On Off] (What's this?)

  • [1] R. Spira, Zero-free regions of $ {\zeta ^{(k)}}(s)$, J. London Math. Soc. 40 (1965), 677-682. MR 31 #5849. MR 0181621 (31:5849)
  • [2] -, Another zero-free region for $ {\zeta ^{(k)}}(s)$, Proc. Amer. Math. Soc. 26 (1970), 246-247. MR 41 #8354. MR 0263754 (41:8354)
  • [3] -, On the Riemann zeta function, J. London Math. Soc. 44 (1969), 325-328. MR 41 #5308. MR 0260684 (41:5308)
  • [4] -, Zeros of $ \zeta (s)$ and the Riemann hypothesis, Illinois J. Math. (to appear).
  • [5] B. Berndt, The number of zeros for $ {\zeta ^{(k)}}(s)$, J. London Math. Soc. (2) 2(1970), 577-580. MR 42 #1776. MR 0266874 (42:1776)
  • [6] H. Bohr and E. Landau, Ein Satz über Dirichletsche Reihen mit Anwendung auf die $ \zeta $-Funktion und die L-Funktionen, Rend. Circ. Mat. Palermo 37 (1914), 269-272.
  • [7] E. C. Titchmarsh, Theory of functions, Oxford Univ. Press, Oxford, 1950.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10H05

Retrieve articles in all journals with MSC: 10H05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0296035-5
Keywords: Riemann zeta function
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society