zeros of Dirichlet eta function
As stated in the parent entry (http://planetmath.org/AnalyticContinuationOfRiemannZetaToCriticalStrip), the definition of the Riemann zeta function may be analytically continued (http://planetmath.org/AnalyticContinuation) from the half-plane to the half-plane by using the Dirichlet eta function via the equation
Then only the status of the points
which are the zeros of , remains : are they poles of or not? E. Landau has 1909 signaled this problem, which has been elementarily solved not earlier than after 40 years, by D. V. Widder. He proved that those numbers, except , are also zeros of . This means that they only are removable singularities of and that (1) in fact extends to every points of the half-plane except .
A new direct proof by J. Sondow of the vanishing of the Dirichlet eta function at the points was published in 2003. It is based on a relation between the partial sums and of the series defining respectively the functions and for , which involves the approximation of an integral by a Riemann sum.
With some clever but not so complicated performed on finite sums, Sondow writes for any the following:
Now if is real, , and , then the factor multiplying is zero and consequently
where denotes a special Riemann sum approximating the integral of over . For , i.e. , one gets
and otherwise, when , one has , giving
Note. By (1) the Dirichlet eta function has as zeros also the zeros of the Riemann zeta function (see Riemann hypothesis (http://planetmath.org/RiemannZetaFunction)).
- 1 E. Landau: Handbuch der Lehre von der Verteilung der Primzahlen. Erster Band. Berlin (1909); p. 161, 933.
- 2 D. V. Widder: The Laplace transform. Princeton University Press (1946); p. 230.
- 3 J. Sondow: “Zeros of the alternating zeta function on the line ”. — Amer. Math. Monthly 110 (2003). Also available http://arxiv.org/abs/math.NT/0209393here.
- 4 J. Sondow: “The Riemann hypothesis, simple zeros, and the asymptotic convergence degree of improper Riemann sums”. — Proc. Amer. Math. Soc. 126 (1998). Also available http://www.ams.org/journals/proc/1998-126-05/S0002-9939-98-04607-3/here.
|Title||zeros of Dirichlet eta function|
|Date of creation||2014-11-21 21:17:02|
|Last modified on||2014-11-21 21:17:02|
|Last modified by||pahio (2872)|