Fact-checked by Grok 2 weeks ago

Dirichlet eta function

The Dirichlet eta function, also known as the alternating zeta function, denoted \eta(s), is a special function in defined by the \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} for complex numbers s with real part \operatorname{Re}(s) > 0. This series converges locally uniformly in the half-plane \operatorname{Re}(s) > 0. It is intimately related to the \zeta(s) through the identity \eta(s) = (1 - 2^{1-s}) \zeta(s), which holds for \operatorname{Re}(s) > 1 and extends by . This relation provides a means to analytically continue \zeta(s) to the region \operatorname{Re}(s) > 0 (excluding the pole at s=1), as \eta(s) remains holomorphic there. The function \eta(s) admits a holomorphic to the entire , with \eta(1) = \ln 2. The non-trivial zeros of \eta(s)—those lying in the critical $0 < \operatorname{Re}(s) < 1—coincide exactly with the non-trivial zeros of \zeta(s), due to the factor $1 - 2^{1-s} having no zeros in this region. This equivalence makes \eta(s) a valuable alternative for studying the Riemann hypothesis, which posits that all such zeros have real part $1/2. Its values at positive integers yield notable constants, such as \eta(2) = \frac{\pi^2}{12} and \eta(3) = \frac{3}{4} \zeta(3), highlighting its role in evaluating series and integrals in number theory and beyond.

Definition and properties

Definition

The Dirichlet eta function, denoted by \eta(s), is defined for complex numbers s with real part \operatorname{Re}(s) > 0 by the Dirichlet series \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}. This series converges in the half-plane \operatorname{Re}(s) > 0. Specifically, it converges absolutely for \operatorname{Re}(s) > 1, since the absolute values yield the Riemann zeta series \sum 1/n^{\operatorname{Re}(s)}, which converges in that region, and converges conditionally for $0 < \operatorname{Re}(s) \leq 1 by the Dirichlet test for convergence applied to the partial sums of the alternating signs. The function was introduced by Peter Gustav Lejeune Dirichlet in 1837 in his seminal paper on the distribution of primes in arithmetic progressions, where it served as an alternating analogue to the Riemann zeta function to facilitate analytic proofs in number theory. It arises naturally in the study of alternating series and forms a special case of the more general Dirichlet L-functions associated with non-principal characters. The eta function is closely related to the Riemann zeta function \zeta(s).

Relation to Riemann zeta function

The Dirichlet eta function \eta(s) is connected to the Riemann zeta function \zeta(s) by the equation \eta(s) = (1 - 2^{1-s}) \zeta(s), which holds for all complex s with \operatorname{Re}(s) > 1, where both are initially defined via their respective .$$] To derive this relation, start with the series for \zeta(s) = \sum_{n=1}^\infty n^{-s}. The subsum over even indices is \sum_{n=1}^\infty (2n)^{-s} = 2^{-s} \zeta(s). The eta series \eta(s) = \sum_{n=1}^\infty (-1)^{n+1} n^{-s} then equals the full zeta series minus twice the even terms, yielding \eta(s) = \zeta(s) - 2 \cdot 2^{-s} \zeta(s) = (1 - 2^{1-s}) \zeta(s).[$$ This identity has significant implications for convergence. While the zeta series diverges for \operatorname{Re}(s) \leq 1, the alternating nature of the eta series ensures convergence for \operatorname{Re}(s) > 0 by the . At s=1, where \zeta(s) has a simple pole, the prefactor $1 - 2^{1-s} vanishes with a simple zero, rendering \eta(1) finite and equal to the alternating harmonic series sum \sum_{n=1}^\infty (-1)^{n+1} n^{-1} = \ln 2 \approx 0.693147.$$] The relation facilitates the of \zeta(s) to the region \operatorname{Re}(s) > 0. Since \eta(s) is holomorphic there (with no singularities), and the factor $1 - 2^{1-s} is holomorphic and nonzero except at the s=1, one obtains \zeta(s) = \eta(s) / (1 - 2^{1-s}) as the meromorphic continuation of the function to this half-plane, with the simple at s=1 arising from the zero in the denominator.[$$

Basic properties

The Dirichlet eta function satisfies the symmetry property \eta(s) = (1 - 2^{1-s}) \zeta(s) for \operatorname{Re}(s) > 0, where \zeta(s) is the Riemann zeta function. This relation arises from separating the even and odd terms in the defining series for \zeta(s), effectively reflecting the contribution of even powers through the factor $2^{1-s}. A fundamental growth estimate for the eta function in its domain of convergence is given by |\eta(s)| \leq \zeta(\operatorname{Re}(s)) for \operatorname{Re}(s) > 0. This bound follows directly from the applied to the alternating defining \eta(s), as the absolute value of each term is at most n^{-\sigma} where \sigma = \operatorname{Re}(s), yielding the partial sum bounded by the corresponding zeta series. For real s > 0, an early identity representing the eta function is \eta(s) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}}{e^t + 1} \, dt. This representation, akin to Hermite's approaches for related zeta functions, provides an alternative expression valid throughout the half-plane \operatorname{Re}(s) > 0 and facilitates numerical evaluation and for positive real arguments.

Analytic continuation

Analytic continuation

The analytic continuation of the Dirichlet eta function to the entire was first established by in his 1859 memoir on the distribution of prime numbers, where he utilized the alternating Dirichlet series to extend the domain beyond its initial convergence region of Re(s) > 0. The primary method for obtaining the meromorphic continuation relies on the fundamental relation between the eta function and the Riemann zeta function: \eta(s) = (1 - 2^{1-s}) \zeta(s). This identity holds initially for Re(s) > 1, but since the zeta function admits an analytic continuation to a meromorphic function on the complex plane with a single simple pole at s = 1, the prefactor (1 - 2^{1-s}) provides the necessary extension for eta. Specifically, the zero of (1 - 2^{1-s}) at s = 1 exactly cancels the residue of the pole in ζ(s), ensuring no singularity remains. As a result, the eta function is holomorphic at s = 1 and, in fact, entire on the whole , with no poles whatsoever. This uniqueness follows directly from the for , as the relation maps the known meromorphic structure of ζ(s) to a holomorphic one for η(s). On the critical line Re(s) = 1/2, the eta function displays controlled asymptotic growth, bounded by |η(1/2 + it)| ≪ t^{1/4} for sufficiently large t > 0, reflecting the subpolynomial increase typical of functions in the critical strip.

Functional equation

The functional equation of the Dirichlet eta function is \eta(1 - s) = \frac{1 - 2^{s}}{1 - 2^{1-s}} 2^{1 - s} \pi^{-s} \Gamma(s) \cos\left( \frac{\pi s}{2} \right) \eta(s), derived from the of the Riemann zeta function via the relation \eta(s) = (1 - 2^{1-s}) \zeta(s). The factor \Gamma(s) is the , providing the interpolating extension of the to the and ensuring the equation holds across the of \eta(s). The term \cos(\pi s / 2) in this form reflects the connection to the zeta function, while the alternating nature is incorporated through the relation. This equation establishes symmetry for \eta(s) around the critical line \operatorname{Re}(s) = 1/2, linking values at s and $1 - s and thus connecting the upper and lower half-planes; if \eta(s) \neq 0, then \eta(1 - s) \neq 0, with implications for the placement of zeros. Unlike the zeta function's functional equation, which features \cos(\pi s / 2) and relates regions affected by the pole at s=1, the eta version remains valid without pole cancellation, as \eta(s) is entire, with the prefactor adjusting for the alternating series.

Particular values

At positive integers

The Dirichlet eta function evaluated at positive integers n \geq 1 is given by the series \eta(n) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^n}, which converges for all such n due to the alternating signs and decreasing terms. At n=1, this reduces to the alternating series, whose sum is the natural logarithm of 2: \eta(1) = \ln 2 \approx 0.693147. For n > 1, the values can be computed using the relation \eta(n) = (1 - 2^{1-n}) \zeta(n), where \zeta(n) is the at positive integers. This connection also expresses \eta(n) in terms of the function as \eta(n) = -\mathrm{Li}_n(-1). At even positive integers n = 2k with k \geq 1, closed-form expressions are available via the known values of \zeta(2k). Specifically, \eta(2k) = (1 - 2^{1-2k}) \zeta(2k) = (-1)^{k+1} \frac{B_{2k} (2\pi)^{2k}}{2 (2k)!}, where B_{2k} are the Bernoulli numbers. For example, \eta(2) = \frac{\pi^2}{12} and \eta(4) = \frac{7\pi^4}{720}. At odd positive integers n = 2k+1 > 1, no simple closed forms in terms of elementary constants like \pi or \ln 2 are known, unlike the even case; instead, values are typically expressed as rational multiples of \zeta(2k+1) or computed numerically from the series. For instance, \eta(3) = \frac{3}{4} \zeta(3) \approx 0.901542. The series for \eta(n) converges absolutely for n > 1, with the rate determined by the decay of $1/k^n, which accelerates as n increases regardless of parity; however, the availability of exact formulas for even n allows precise evaluation without summation, while odd n generally require numerical approximation despite similar convergence behavior.

At negative integers

The values of the Dirichlet eta function at negative integers are obtained via its analytic continuation, which relates it directly to the Riemann zeta function. Specifically, for a positive integer n, \eta(-n) = (1 - 2^{1+n}) \zeta(-n), where \zeta(s) is the Riemann zeta function. This relation yields rational numbers for \eta(-n), as \zeta(-n) itself takes rational values at these points. Representative examples illustrate this: \eta(-1) = \frac{1}{4}, \eta(-2) = 0, and \eta(-3) = -\frac{1}{8}. A pattern emerges based on parity: at negative even integers -2k (for k \geq 1), \eta(-2k) = 0 because \zeta(-2k) = 0; at negative odd integers, the values are nonzero rationals that alternate in sign while growing in magnitude. These rational values connect to Bernoulli numbers through the known expression for the zeta function at negative integers: for nonnegative integer m, \zeta(-m) = (-1)^m \frac{B_{m+1}}{m+1}, where B_j are the Bernoulli numbers (with the convention B_1 = -\frac{1}{2}). Substituting into the relation for \eta gives explicit rational expressions in terms of Bernoulli numbers; for instance, at negative odd integers s = 1 - 2k (with k \geq 1), \eta(1 - 2k) = \frac{2^{2k} - 1}{2k} B_{2k}. This formula highlights the role of even-indexed Bernoulli numbers in determining the eta values at these points. In physical contexts, such as , the eta function at negative integers provides finite results via regularization of divergent sums, notably in calculations of the for between boundaries.

At s=0 and s=1

The Dirichlet eta function evaluated at s = 1 yields \eta(1) = \ln 2, approximately 0.693147, representing the sum of the alternating harmonic series \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}, which converges conditionally. This finite value at s = 1 is significant because the related \zeta(s) has a simple pole there with residue 1, but the prefactor $1 - 2^{1-s} in the relation \eta(s) = (1 - 2^{1-s}) \zeta(s) vanishes exactly at this point, canceling the singularity and ensuring analyticity. The result connects to broader , where the provides a convergent counterpart to the divergent ordinary harmonic series, indirectly linking to through asymptotic expansions of partial sums. At s = 0, the defining Dirichlet series diverges, but analytic continuation gives \eta(0) = \frac{1}{2}. This value follows from the relation \eta(s) = (1 - 2^{1-s}) \zeta(s), substituting \zeta(0) = -\frac{1}{2} to obtain (1 - 2) \left( -\frac{1}{2} \right) = \frac{1}{2}. Numerical confirmation comes from the limit \lim_{s \to 0^+} \eta(s) = \frac{1}{2}, consistent with the function's behavior near the boundary of convergence. The functional equation further supports this evaluation, reflecting the eta function's meromorphic extension to the entire complex plane except for the pole inherited from zeta at s = 1. This special value at s = 0 interprets the formal alternating sum of units (Grandi's series) via regularization methods like Cesàro summation, which also yields \frac{1}{2}, aligning the analytic result with summation theory.

Zeros

Trivial zeros

The trivial zeros of the Dirichlet eta function \eta(s) are located at the negative even integers s = -2k for each positive integer k = 1, 2, 3, \dots, that is, at s = -2, -4, -6, \dots. These zeros arise because \eta(s) = (1 - 2^{1-s}) \zeta(s), where \zeta(s) is the , and \zeta(-2k) = 0 while the prefactor $1 - 2^{1 - (-2k)} = 1 - 2^{1 + 2k} \neq 0. The vanishing of \zeta(-2k) follows from the explicit formula \zeta(-n) = -B_{n+1}/(n+1) for positive integers n, where the numbers satisfy B_{m} = 0 for all odd m > 1; thus, for n = 2k even, B_{2k+1} = 0 implies \zeta(-2k) = 0. An alternative perspective on these zeros comes from the of the eta function, \eta(s) = \lambda(s) \eta(1 - s), where \lambda(s) = \frac{1 - 2^{1-s}}{1 - 2^{s}} \cdot (2^{s} \pi^{s-1}) \sin(\pi s / 2) \Gamma(1 - s). At s = -2k, the factor \sin(\pi s / 2) = \sin(-\pi k) = 0, while \eta(1 - (-2k)) = \eta(1 + 2k) is finite and nonzero (as $1 + 2k is a positive odd integer greater than 1), and the remaining factors in \lambda(-2k) are also finite and nonzero. Consequently, the equation forces \eta(-2k) = 0, inheriting the zero from the sine factor. All trivial zeros of \eta(s) are simple, with multiplicity one, because the sine function has simple zeros at these points, and the other components of the do not introduce additional factors of vanishing or poles at s = -2k. Unlike the , which is meromorphic with a single simple pole at s = 1, the eta function is an with no poles anywhere in the . Thus, its trivial zeros at the negative even integers are genuine zeros without any associated residues or singularities, reflecting the alternating nature of its that ensures convergence and to the full plane.

Non-trivial zeros

The non-trivial zeros of the Dirichlet eta function \eta(s) are those lying in the critical strip $0 < \Re(s) < 1. These zeros coincide exactly with the non-trivial zeros of the Riemann zeta function \zeta(s), since \eta(s) = (1 - 2^{1-s}) \zeta(s) and the prefactor $1 - 2^{1-s} has no zeros in this region. The functional equation of \eta(s) implies that the non-trivial zeros are symmetric with respect to both the real axis and the critical line \Re(s) = 1/2. The Riemann hypothesis analog for the eta function states that all such zeros satisfy \Re(s) = 1/2; this conjecture is equivalent to the classical Riemann hypothesis for \zeta(s), given the shared zeros. The first non-trivial zero of \eta(s) occurs at s \approx 1/2 + 14.1347251417 i, with its complex conjugate below the real axis; subsequent zeros appear at approximately $1/2 + 21.0220396388 i and $1/2 + 25.0108575801 i. Extensive numerical computations have confirmed that the first $10^{13} non-trivial zeros all lie on the critical line \Re(s) = 1/2. Zero-free regions for the non-trivial zeros of \eta(s) exist near \Re(s) = 1, mirroring those for \zeta(s). For instance, there are no zeros in the region \Re(s) > 1 - \frac{c}{(\log | \Im(s) |)^{2/3} (\log \log | \Im(s) |)^{1/3}} for sufficiently large | \Im(s) | and a suitable constant c > 0, with recent improvements providing explicit bounds.

Representations

Dirichlet series

The Dirichlet eta function is defined by the following Dirichlet series, which converges absolutely for \Re(s) > 1 and conditionally for $0 < \Re(s) \leq 1: \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}. The coefficients alternate in sign based on the parity of n. The eta function is closely related to the Riemann zeta function \zeta(s) = \sum_{n=1}^{\infty} n^{-s} via the identity \eta(s) = (1 - 2^{1-s}) \zeta(s), which follows directly from separating the terms in the zeta series into even and odd indices and scaling the even terms by -2 \cdot 2^{-s}. This relation highlights eta as a specific instance of a Dirichlet series with periodic coefficients of period 2. The eta function is related to the Dirichlet L-function L(s, \chi_4) associated with the non-principal character \chi_4 modulo 4, up to a simple entire factor. For improved numerical convergence, particularly in the strip $0 < \Re(s) \leq 1 where the original series converges slowly, accelerated expansions can be derived using the Euler-Maclaurin summation formula applied to the partial sums of the alternating series. These accelerated forms are particularly useful for high-precision computations, outperforming the naive series by incorporating asymptotic expansions of the tail.

Integral representations

The Dirichlet eta function admits a Mellin transform representation valid for \Re(s) > 0: \eta(s) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}}{e^t + 1}\, dt. This formula arises from the expansion \frac{1}{e^t + 1} = \sum_{n=1}^\infty (-1)^{n-1} e^{-nt}, which converges absolutely for t > 0, followed by term-by-term integration justified by the , yielding the defining for \eta(s). The integral converges in this half-plane due to the behavior of the integrand near t = 0 and t = \infty. This representation facilitates to the (except for a simple pole at s = 1) by deforming the integration path to a Hankel contour H that encircles the branch cut along the negative real axis in the positive (counterclockwise) direction, starting and ending at +\infty: \eta(s) = \frac{1}{2\pi i \Gamma(s)} \int_H \frac{(-t)^{s-1}}{e^t + 1}\, dt, where (-t)^{s-1} = e^{(s-1) \log(-t)} with branch of the logarithm (argument from -\pi to \pi). The contour avoids the origin and the cut, capturing residues at poles of the integrand to extend beyond \Re(s) > 0. The eta function is also linked to Fourier series expansions of periodic functions with discontinuities, such as the sawtooth wave f(x) = (\pi - x)/2 for $0 < x < 2\pi, whose Fourier series is \sum_{n=1}^\infty \sin(nx)/n. Integrating or differentiating such series term by term generates alternating sums that evaluate to particular values of \eta(s); for example, repeated integration yields \eta(2k+1) in terms of Bernoulli numbers and powers of \pi. This connection provides explicit evaluations at positive odd integers and highlights the role of eta in representing Fourier coefficients of piecewise linear functions. A Herglotz-type integral representation expresses the real part of \eta(s) for \sigma = \Re(s) > 0 as a positive : \Re \eta(\sigma + it) = \frac{1}{\pi} \int_{-\infty}^\infty \frac{y}{(x - \sigma)^2 + (t - y)^2} \log \left( \frac{\pi}{\sinh \pi y} \right) dy + c, where c is a constant depending on the boundary values; this follows from the Herglotz theorem applied to the for the upper half-plane, using the known boundary behavior of \eta(it) related to the logarithm of the . This form emphasizes the positivity and growth properties of \Re \eta(s) in the critical strip.

Numerical methods

General algorithms

The Dirichlet eta function η(s) can be evaluated numerically for complex s with Re(s) > 0 using direct summation of its defining : \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}. This series converges conditionally by , as the partial sums of the signs (-1)^{n-1} are bounded and 1/n^σ decreases monotonically to zero for σ = Re(s) > 0. For practical computation, the after N terms is bounded by the first omitted term's , |(N+1)^{-s}|, leveraging the estimation theorem. This method is straightforward but requires a large number of terms for accuracy in regions of slow . For Re(s) > 1, a more efficient approach computes η(s) via its relation to the ζ(s): \eta(s) = (1 - 2^{1-s}) \zeta(s). Here, ζ(s) is evaluated using faster-converging methods like the Euler-Maclaurin formula on its non-alternating series, avoiding the conditional convergence issues of the eta series in this half-plane. This transformation is particularly useful since the factor (1 - 2^{1-s}) is easily computed and non-zero. On the critical line Re(s) = 1/2, η(s) is computed by first applying the Riemann-Siegel formula to obtain ζ(1/2 + it) for large t = Im(s), which provides an involving a main up to √t and a remainder term of similar length, achieving O(t^{1/4 + ε}) . The eta value is then obtained by multiplying by (1 - 2^{1-s}), inheriting the efficiency of the zeta computation while benefiting from the eta function's better-behaved series for verification. This adaptation is essential for exploring zeros or high-precision values at large heights. Software libraries provide robust implementations for general numerical evaluation. In Mathematica, DirichletEta supports , internally switching between series , the relation, and functional equation-based methods depending on s and requested digits. The mpmath library computes η(s) via mpmath.dirichlet(s, [-1, 1]), which handles high precision (up to thousands of digits) by employing the relation for Re(s) > 0 and elsewhere, with users specifying working precision to manage rounding errors in oscillatory regions. These tools ensure reliable results but require careful precision settings for s near the line Re(s) = 0. Challenges in numerical computation include slow convergence of the direct series near s = 1, where hundreds or thousands of terms may be needed due to the nearby pole of ζ(s), despite η(1) = ln(2) being finite. For large |Im(s)|, the terms oscillate rapidly, increasing computational cost unless accelerated; integral representations offer one avenue for improvement in such cases.

Borwein's method

Borwein's method, developed by Peter Borwein in the late 1990s and published in 2000, provides an efficient numerical algorithm for computing the Dirichlet eta function η(s) at positive integers s, leveraging the relationship η(s) = (1 - 2^{1-s}) ζ(s) with the Riemann zeta function and accelerating the alternating Dirichlet series through polynomial approximations. The approach draws parallels to acceleration techniques for ζ(s) and employs a specific recursive sequence for exponential convergence. This method is grounded in an integral representation of η(s) and is particularly useful for high-precision calculations at integer arguments, building on earlier work in series acceleration for L-functions. The algorithm begins with the integral form derived from the : \Gamma(s) \eta(s) = \int_0^\infty \frac{t^{s-1}}{e^t + 1} \, dt, which can be transformed via substitution x = e^{-t} to \eta(s) = \frac{1}{\Gamma(s)} \int_0^1 \frac{(-\ln x)^{s-1}}{1 + x} \, dx. To accelerate convergence, the method uses a defined by d_0 = 1, and for k \geq 1, d_k = k \sum_{\ell=0}^k \frac{(k + \ell - 1)! \, 4^\ell}{(k - \ell)! \, (2\ell)!}, though the explicit form is approximated via the sum \eta(s) = -\frac{1}{d_n} \sum_{k=0}^{n-1} (-1)^k \frac{d_k - d_n}{(k+1)^s} + R_n(s), with the remainder term R_n(s) satisfying |R_n(s)| \leq \frac{3 (1 + 2 |\Im(s)|)}{(3 + \sqrt{8})^n} \exp\left(\frac{\pi}{2} |\Im(s)|\right) for \Re(s) \geq 1/2, providing exponential convergence as n increases. The process iterates by increasing n to refine the approximation until the desired precision is reached. This method offers significant advantages over direct summation of the \sum_{k=1}^\infty (-1)^{k-1} / k^s, which converges linearly; for large positive integers s = 2k, the acceleration requires fewer terms to attain high precision, scaling better for extended computations despite the direct series already being fast. For small k, the values are known exactly via closed forms, allowing verification, but excels in uniform efficiency across integer arguments without relying on special cases. Historically, it extends Borweins' earlier work on zeta function evaluation and pi algorithms, emphasizing practical high-precision arithmetic. As an example, for η(4), the direct closed form is η(4) = (7/8) ζ(4) = 7 π^4 / 720 ≈ 0.9470328294, but applying Borwein's method with a low n (e.g., n=4) yields an with error less than 10^{-16}, converging rapidly to the exact value upon iteration; this demonstrates the method's ability to produce arbitrary precision without invoking the formula for ζ(4) = π^4 / 90.

Derivatives and advanced topics

Derivatives

The first derivative of the Dirichlet eta function \eta(s) can be obtained by termwise of its defining \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}, yielding \eta'(s) = -\sum_{n=1}^\infty \frac{(-1)^{n-1} \ln n}{n^s} for \operatorname{Re}(s) > 0, where the result holds by from the region of \operatorname{Re}(s) > 1. Alternatively, using the relation \eta(s) = (1 - 2^{1-s}) \zeta(s) to the Riemann zeta function \zeta(s), differentiation gives the expression \eta'(s) = 2^{1-s} (\ln 2) \zeta(s) + (1 - 2^{1-s}) \zeta'(s), which facilitates computation and analysis across the complex plane. Higher-order derivatives follow from repeated application of the Leibniz product rule to the relation with \zeta(s), or directly from the series as \eta^{(k)}(s) = (-1)^k \sum_{n=1}^\infty \frac{(-1)^{n-1} (\ln n)^k}{n^s} for k \geq 1 and \operatorname{Re}(s) > 0, again justified by analytic continuation. Evaluating at specific points, such as s = 0, yields \eta'(0) = \frac{1}{2} \ln\left(\frac{\pi}{2}\right), which can be derived by differentiating the of \eta(s) or substituting known values \zeta(0) = -\frac{1}{2} and \zeta'(0) = -\frac{1}{2} \ln(2\pi) into the first-derivative expression. These derivatives play a role in applications such as the study of analogs to the Stieltjes constants for \eta(s), defined via the expansion around s=1 where \eta(1) = \ln 2, providing coefficients that extend properties of the function's .

Landau's problem and related conjectures

In 1912, Edmund Landau posed the question of whether the Riemann zeta function \zeta(1/2 + it) changes sign infinitely often for t > 0. This is equivalent to the existence of infinitely many non-trivial zeros on the critical line \operatorname{Re}(s) = 1/2. The problem was affirmatively resolved by G. H. Hardy in 1914, who proved that \zeta(s) has infinitely many zeros on this line. The Dirichlet eta function provides an equivalent reformulation through the relation \eta(s) = (1 - 2^{1-s}) \zeta(s), which holds for \operatorname{Re}(s) > 1 and extends analytically to \operatorname{Re}(s) > 0 except at s=1, where the pole of \zeta(s) is canceled by the zero of the prefactor. The zeros of \eta(s) in the critical strip $0 < \operatorname{Re}(s) < 1 coincide exactly with those of \zeta(s), as the prefactor $1 - 2^{1-s} has no zeros there. Moreover, the alternating series defining \eta(s) converges conditionally for $0 < \operatorname{Re}(s) \leq 1, enabling more effective numerical evaluation and analytic study of the sign changes of \zeta(1/2 + it) compared to the non-alternating zeta series. The first non-trivial zero of both functions, marking the initial sign change, occurs at s \approx 1/2 + 14.134725 i. Although infinitely many zeros lie on the critical line, determining the precise proportion remains open, tied to the Riemann hypothesis that all non-trivial zeros do. Partial progress includes N. Levinson's 1974 result showing that more than one third of the zeros up to height T lie on the line as T \to \infty. This was improved by H. M. Bui, J. B. Conrey, and M. P. Young in 2011 to more than 41%, using refined methods and moment estimates; this bound remains the strongest as of 2025. The alternating structure of \eta(s) has aided such density proofs by allowing better control over error terms in the critical strip.90023-9) Related open conjectures concern the fine-scale distribution of these zeros. Montgomery's pair correlation conjecture (1973) posits that the pairwise spacings of normalized zeros on the critical line follow the pair correlation measure of the Gaussian Unitary Ensemble from random matrix theory, implying strong forms of the . The eta function's rapid convergence on the line has facilitated partial verifications and extensions of pair correlation results, particularly in computing high moments and correlations near low-lying zeros.