zeros of Dirichlet eta function

As stated in the parent entry (, the definition of the Riemann zeta functionDlmfDlmfMathworldPlanetmath may be analytically continued ( from the half-plane  s>1  to the half-plane  s>0  by using the Dirichlet eta functionMathworldPlanetmath η(s) via the equation

ζ(s)=η(s)1-22s. (1)

Then only the status of the points

sn:= 1+n2πiln2  (n) (2)

which are the zeros of 1-22s, remains :  are they poles of ζ(s) 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  s=1,  are also zeros of η(s).  This means that they only are removable singularities of ζ(s) and that (1) in fact extends ζ(s) to every points of the half-plane  s>0  except  s=1.

A new direct proof by J. Sondow of the vanishing of the Dirichlet eta function at the points  sn1  was published in 2003.  It is based on a relation between the partial sums ηn(s) and ζn(s) of the series defining respectively the functionsMathworldPlanetmath η(s) and ζ(s) for s>1, which involves the approximation of an integralDlmfPlanetmath by a Riemann sum.

With some clever but not so complicated performed on finite sums, Sondow writes for any s the following:

η2n(s) = 1-12s+13s-14s+-+(-1)2n-1(2n)s
= 1+12s+13s+14s++(-1)2n-1(2n)s-2(12s+14s++1(2n)s)

Now if t is real,  s=1+it,  and  21-s=2-it=1,  then the factor multiplying ζ2n(s) is zero and consequently


where  Rn(f(x),a,b)  denotes a special Riemann sum approximating the integral of f(x) over  [a,b].  For  s=1,  i.e.  t=0, one gets


and otherwise, when  t0,  one has  |n1-s|=|n-it|=1,  giving

|η(s)| =limn|η2n(s)|=limn|Rn(1/(1+x)s,0,1)|
=|01dx(1+x)s|=|21-s-11-s|=|1-1-it|= 0.

Note.  By (1) the Dirichlet eta function has as zeros also the zeros of the Riemann zeta function (see Riemann hypothesis (


  • 1 E. Landau: Handbuch der Lehre von der Verteilung der Primzahlen. Erster Band. Berlin (1909); p. 161, 933.
  • 2 D. V. Widder: The Laplace transformDlmfMathworldPlanetmath.  Princeton University Press (1946); p. 230.
  • 3 J. Sondow: “Zeros of the alternating zeta function on the line  s=1”.  — Amer. Math. Monthly 110 (2003).  Also available
  • 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
Title zeros of Dirichlet eta function
Canonical name ZerosOfDirichletEtaFunction
Date of creation 2014-11-21 21:17:02
Last modified on 2014-11-21 21:17:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Derivation
Classification msc 30D30
Classification msc 30B40
Classification msc 11M41
Related topic DirichletEtaFunction