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

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 *(s

Both the Dirichlet eta function and the Riemann zeta function are special cases of polylogarithms.

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.

represents the gamma function). This gives the eta function as a Mellin transform.

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

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.

Zeros

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 where n is any nonzero integer.

The zeros of the eta function are located symmetrically with respect to the real axis and under the Riemann hypothesis, and on the perpendicular half line formed by the negative real axis.

Landau's problem

In the equation , "the pole of at is cancelled by the zero of the other factor" (Titchmarsh, 1986, p.17), and as a result is neither infinite nor zero (see η must be zero at all the points , 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, 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 and with some zeros also on , but not equal to those of eta.

.

Integral representations

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

The Cauchy–Schlömilch transformation (Amdeberhan, Moll et al., 2010) can be used to prove this other representation, valid for . Integration by parts of the first integral above in this section yields another derivation.

This corresponds to a Jensen

Numerical algorithms

Most of the series acceleration techniques developed for alternating series can be profitably applied to the evaluation of the eta function. One particularly simple, yet reasonable method is to apply Euler's transformation of alternating series

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

Borwein's method

Peter Borwein used approximations involving Chebyshev polynomials to produce a method for efficient evaluation of the eta function. the error term in the error bound indicates that the Borwein series converges quite rapidly as n increases.

This form appears similar to the definition of the eta function but with finitely many terms, with each term being multiplied by a scaling factor.

Particular values

• η(0) = , the Abel sum of Grandi's series 1 − 1 + 1 − 1 + ....

• η(−1) = , the Abel sum of 1 − 2 + 3 − 4 + ⋯.

• For positive integer k where is the kth Bernoulli number.

Also:

• , this is the alternating harmonic series

Taking the limit , one obtains .

Derivatives

The derivative with respect to the parameter

References

• Landau, Edmund, Handbuch der Lehre von der Verteilung der Primzahlen, Erster Band, Berlin, 1909, p.160. (Second edition by Chelsea, New York, 1953, p.160, 933)

• Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta Function, Second revised (Heath-Brown) edition. Oxford University Press.

• Borwein, P., An Efficient Algorithm for the Riemann Zeta Function , Constructive experimental and nonlinear analysis, CMS Conference Proc. 27 (2000), 29–34.

• Amer. Math. Monthly 112 (2005) 61–65, formula 18.

• Amer. Math. Monthly, 110 (2003) 435–437.

• p.12.

• .