Dirichlet eta function

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:

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).

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.

, although in fact these hypothetical additional poles do not exist.)

which is also defined in the region of positive real part (

The zeros of the eta function include all the zeros of the zeta function: the negative even integers (real equidistant simple zeros); the 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.

adds an infinite number of complex simple zeros, located at equidistant points on the line

The zeros of the eta function are located symmetrically with respect to the real axis and under the Riemann hypothesis would be on two parallel lines

, and on the perpendicular half line formed by the negative real axis.

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 (see § Particular values).

, 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

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

is real and strictly positive, the series converges since the regrouped terms alternate in sign and decrease in absolute value to zero.

According to a theorem on uniform convergence of Dirichlet series first proven by Cahen in 1894, the

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

With some simple algebra performed on finite sums, we can write for any complex s

where Rn(f(x), a, b) denotes a special Riemann sum approximating the integral of f(x) over [a, b].

A number of integral formulas involving the eta function can be listed.

The first one follows from a change of variable of the integral representation of the Gamma function (Abel, 1823), giving a Mellin transform which can be expressed in different ways as a double integral (Sondow, 2005).

The Cauchy–Schlömilch transformation (Amdeberhan, Moll et al., 2010) can be used to prove this other representation, valid for

The next formula, due to Lindelöf (1905), is valid over the whole complex plane, when the principal value is taken for the logarithm implicit in the exponential.

"This formula, remarquable by its simplicity, can be proven easily with the help of Cauchy's theorem, so important for the summation of series" wrote Jensen (1895).

Similarly by converting the integration paths to contour integrals one can obtain other formulas for the eta function, such as this generalisation (Milgram, 2013) valid for

The zeros on the negative real axis are factored out cleanly by making

One particularly simple, yet reasonable method is to apply Euler's transformation of alternating series, to obtain

Note that the second, inside summation is a forward difference.

Peter Borwein used approximations involving Chebyshev polynomials to produce a method for efficient evaluation of the eta function.

in the error bound indicates that the Borwein series converges quite rapidly as n increases.