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Dirichlet eta function

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Title: Dirichlet eta function  
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Subject: Riemann zeta function, Eta function, WikiProject Mathematics/PlanetMath Exchange/00-XX General, ETA (separatist group), Occurrences of Grandi's series
Collection: Zeta and L-Functions
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Dirichlet eta function

Color representation of the Dirichlet eta function. It is generated as a Matplotlib plot using a version of the Domain coloring method.[1]

In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0:

\eta(s) = \sum_{n=1}^{\infty}{(-1)^{n-1} \over n^s} = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots

This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s). The following simple relation holds:

\eta(s) = \left(1-2^{1-s}\right) \zeta(s)

While the Dirichlet series expansion for the eta function is convergent only for any complex number s with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function (and the above relation then shows the zeta function is meromorphic with a simple pole at s = 1, and perhaps poles at the other zeros of the factor 1-2^{1-s}).

Equivalently, we may begin by defining

\eta(s) = \frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x+1}{dx}

which is also defined in the region of positive real part. This gives the eta function as a Mellin transform.

Hardy gave a simple proof of the functional equation for the eta function, which is

\eta(-s) = 2 \frac{1-2^{-s-1}}{1-2^{-s}} \pi^{-s-1} s \sin\left({\pi s \over 2}\right) \Gamma(s)\eta(s+1).

From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.

Contents

  • Zeros 1
  • Landau's problem with ζ(s) = η(s)/0 and solutions 2
  • Integral representations 3
  • Numerical algorithms 4
    • Borwein's method 4.1
  • Particular values 5
  • Derivatives 6
  • References 7

Zeros

The zeros of the eta function include all the zeros of the zeta function: the infinity of negative even integers (real equidistant simple zeros); an infinity of zeros along the critical line, none of which are known to be multiple and over 40% of which have been proven to be simple, and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of rectangles symmetrical around the x-axis and the critical line and whose multiplicity is unknown. In addition, the factor 1-2^{1-s} adds an infinity of complex simple zeros, located at equidistant points on the line \Re(s)=1, at s_n=1+2n\pi i/\log(2) where n is any nonzero integer.

Under the Riemann hypothesis, the zeros of the eta function would be located symmetrically with respect to the real axis on two parallel lines \Re(s)=1/2, \Re(s)=1, and on the perpendicular half line formed by the negative real axis.

Landau's problem with ζ(s) = η(s)/0 and solutions

In the equation η(s) = (1−21−s) ζ(s), "the pole of ζ(s) at s=1 is cancelled by the zero of the other factor" (Titchmarsh, 1986, p. 17), and as a result η(1) is neither infinite nor zero. However, in the equation

\zeta(s)=\frac{\eta(s)}{1-2^{1-s}},

η must be zero at all the points s_n = 1+n\frac{2\pi}{\ln{2}}i, n\ne0, n \in Z , where the denominator is zero, if the Riemann zeta function is analytic and finite there. The problem of proving this without defining the zeta function first was signaled and left open by E. Landau in his 1909 treatise on number theory: "Whether the eta series is different from zero or not at the points s_n\ne1, i.e., whether these are poles of zeta or not, is not readily apparent here."

A first solution for Landau's problem was published almost 40 years later by D. V. Widder in his book The Laplace Transform. It uses the next prime 3 instead of 2 to define a Dirichlet series similar to the eta function, which we will call the \lambda function, defined for \Re(s)>0 and with some zeros also on \Re(s)=1, but not equal to those of eta.

An elementary direct and \zeta\,-independent proof of the vanishing of the eta function at s_n\ne1 was published by J. Sondow in 2003. It expresses the value of the eta function as the limit of special Riemann sums associated to an integral known to be zero, using a relation between the partial sums of the Dirichlet series defining the eta and zeta functions for \Re(s)>1.

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