The eta function is examined over the critical strip
and there is an investigation of the statement that all zeros of the zeta function must lie on the critical line
. A further investigation is made into the claim that there are no other zeros of the zeta or eta functions within the critical strip.
References
[1]
Dwilewicz, R.J. and Mináč, J. (2009) Values of the Riemann Zeta Function at Integers. Materials Matemàtics, 2009, 26 p.
[2]
Digital Library of Mathematical Functions. https://dlmf.nist.gov/
[3]
The Riemann Zeta Function, the Dirichlet Eta Function. https://en.wikipedia.org/wiki/Dirichlet_eta_function
[4]
Heinbockel, J.H. (2021) Special Values for the Riemann Zeta Function. JournalofAppliedMathematicsandPhysics, 9, 1108-1120.
[5]
Titchmarch, E.C. (1986) The Theory of the Riemann-Zeta Function. 2nd Edition, Revised by D.R. Heath-Brown, The Clarendon Press.
[6]
Sondow, J. (1994) Analytic Continuation of Riemann’s Zeta Function and Values at Negative Integers via Euler’s Transformation of Series. ProceedingsoftheAmericanMathematicalSociety, 120, 421-424. https://doi.org/10.2307/2159877
[7]
Boyadzhiev, K. and Frontczak, R. (2021) Series Involving Euler’s Eta (or Dirichlet Eta) Function. JournalofIntegerSequences, 24, Article 21.9.1.
[8]
Hardy, G.H. and Littlewood, J.E. (1921) The Zeros of Riemann’s Zeta-Function on the Critical Line. MathematischeZeitschrift, 10, 283-317. https://doi.org/10.1007/bf01211614
