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Hurwitz zeta function

The Hurwitz zeta function, denoted \zeta(s, a), is a special function in analytic number theory defined by the Dirichlet series \zeta(s, a) = \sum_{n=0}^{\infty} (n + a)^{-s} for complex numbers s with \operatorname{Re}(s) > 1 and real a > 0.^[1]^[2] It generalizes the Riemann zeta function \zeta(s), recovering it when a = 1.^[1] Introduced by Adolf Hurwitz in 1882, it serves as a foundational tool for studying generalizations of zeta functions and their roles in sums over arithmetic progressions.^[2] This function admits an analytic continuation to a meromorphic function on the entire complex plane, featuring a single simple pole at s = 1 with residue 1, independent of a.^[2]^[1] For integer values, particularly negative integers, it connects directly to Bernoulli polynomials via \zeta(-n, a) = -\frac{B_{n+1}(a)}{n+1} for positive integers n, where B_k(x) are the Bernoulli polynomials.^[1] An integral representation further aids computation and analysis: \zeta(s, a) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}}{e^{a t} (1 - e^{-t})} \, dt for \operatorname{Re}(s) > 1 and \operatorname{Re}(a) > 0.^[1] Key relations link the Hurwitz zeta to other special functions, including the polygamma function through \psi^{(m)}(z) = (-1)^{m+1} m! \zeta(m+1, z) for m \geq 0.^[1] It also underlies Dirichlet L-functions and periodic zeta functions, with a functional equation for rational a = p/q in lowest terms: \zeta(s, p/q) = 2 \Gamma(1-s) (2\pi q)^{s-1} \sum_{n=1}^q \sin\left(\frac{\pi s}{2} + \frac{2\pi n p}{q}\right) \zeta(1-s, n/q).^[1] These properties enable its use in evaluating generalized harmonic numbers and power sums.^[1] In applications, the Hurwitz zeta function appears in number theory for analytic continuations of L-series and moment calculations of Dirichlet L-functions, as well as in physics for computing Casimir energies in quantum field theory via products of zeta values.^[3]^[4] Its monotonicity and convexity properties further support inequalities in analytic estimates.^[5] High-precision algorithms, such as those based on the Euler-Maclaurin formula, facilitate its numerical evaluation with rigorous error bounds.^[6]

Definition and Properties

Definition

The Hurwitz zeta function, denoted \zeta(s, a), is defined for complex numbers s with \operatorname{Re}(s) > 1 and a with \operatorname{Re}(a) > 0 by the Dirichlet series

\zeta(s, a) = \sum_{n=0}^{\infty} (n + a)^{-s}.

This series converges absolutely in the specified half-plane, providing the initial analytic representation of the function.^[2]^[7] The parameter s serves as the complex variable, analogous to that in the Riemann zeta function, while a acts as a shift parameter that generalizes the summation index, often taken to be real and positive for simplicity in many applications. When a = 1, the Hurwitz zeta function reduces to the Riemann zeta function \zeta(s) = \zeta(s, 1), highlighting its role as a natural extension.^[1]^[2] Introduced by Adolf Hurwitz in 1882, the function arose in the study of properties of Dirichlet series related to class numbers of binary quadratic forms.^[1] For fixed s > 1, \zeta(s, a) is strictly decreasing in a > 0, as evidenced by the derivative \frac{\partial}{\partial a} \zeta(s, a) = -s \zeta(s+1, a) < 0. Consequently, $0 < a < 1 implies \zeta(s, a) > \zeta(s, 1).^[1]

Analytic Continuation and Basic Properties

The Hurwitz zeta function \zeta(s, a), initially defined by its Dirichlet series for \operatorname{Re}(s) > 1 and $0 < a \leq 1, admits an analytic continuation to a meromorphic function on the entire complex plane \mathbb{C}. This continuation has a single singularity, a simple pole at s = 1 with residue 1, independent of the parameter a. The meromorphic extension can be constructed using contour integral representations, such as those involving the gamma function, which converge everywhere except at the pole, or via functional equations that relate values at s and $1-s.^[2]^[1] This analytic continuation is unique, as it provides the only meromorphic function in \mathbb{C} that agrees with the original series in the half-plane \operatorname{Re}(s) > 1. For fixed a, the function exhibits controlled growth in vertical strips of the complex plane, reflecting subexponential growth similar to the Riemann zeta function. A fundamental symmetry property arises from the structure of the continuation: \zeta(s, a) + \zeta(s, 1-a) can be expressed in terms of trigonometric sums involving the parameter a, providing a reflection relation that connects values symmetric about a = 1/2. This relation, derived non-constructively from the functional equation, underscores the periodic nature of the function in a modulo 1.^[1]

Representations

Dirichlet Series Representation

The Hurwitz zeta function admits a Dirichlet series representation given by

\zeta(s, a) = \sum_{n=0}^{\infty} \frac{1}{(n + a)^{s}}

for complex numbers s with \operatorname{Re}(s) > 1 and a \in \mathbb{C} \setminus \{0, -1, -2, \dots \}. This series converges absolutely in the half-plane \operatorname{Re}(s) > 1, while conditional convergence holds in a larger region through analytic continuation to the meromorphic function defined on the entire complex plane except for a simple pole at s = 1. To accelerate numerical evaluation of this slowly converging series, particularly for \operatorname{Re}(s) > 0, the Euler-Maclaurin summation formula is applied to approximate the tail beyond a finite sum \sum_{n=0}^{N-1} (n + a)^{-s}. The formula yields

\zeta(s, a) = \sum_{n=0}^{N-1} (n + a)^{-s} + \frac{(N + a)^{1 - s}}{s - 1} + \frac{(N + a)^{-s}}{2} + \sum_{k=1}^{m} \frac{B_{2k}}{(2k)!} (s)_{2k - 1} (N + a)^{1 - s - 2k} + R,

where B_{2k} are Bernoulli numbers, (s)_{2k-1} = \frac{\Gamma(s + 2k - 1)}{\Gamma(s)} is the rising Pochhammer symbol, the sum involves higher-order correction terms derived from derivatives of (n + a)^{-s}, and R is a remainder term controllable by choosing m and N appropriately.^[6] This approach significantly improves convergence speed for high-precision computations, outperforming direct summation for moderate to large N.^[8] For large |a| with fixed s, asymptotic expansions of \zeta(s, a) follow directly from the Euler-Maclaurin formula, providing series in powers of a^{-1}:

\zeta(s, a) \sim a^{-s} + \frac{1}{2} a^{-s-1} + \sum_{k=1}^{\infty} \frac{B_{2k}}{(2k)!} \frac{\Gamma(s + 2k - 1)}{\Gamma(s)} a^{1 - s - 2k},

valid as |a| \to \infty in \operatorname{Re}(s) > 0, with error bounds ensuring rapid convergence for the truncated series.^[2] These expansions are particularly useful for analyzing behavior when the parameter a is dominant, such as in approximations near integer values.^[9] For positive integer orders, the Dirichlet series connects to the polygamma function via \psi^{(m-1)}(a) = (-1)^{m} (m-1)! \zeta(m, a) where m \geq 2, linking the Hurwitz zeta to derivatives of the logarithm of the gamma function. This relation facilitates computations and properties transfer between the functions for integer s.^[10]

Integral Representations

The Hurwitz zeta function admits several integral representations that facilitate its evaluation and analytic continuation beyond the region of absolute convergence of the defining Dirichlet series. One fundamental form is the Mellin transform representation, which arises from interchanging the order of summation and integration in the series definition. For \Re(s) > 1 and \Re(a) > 0,

\zeta(s, a) = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} \frac{e^{-a t}}{1 - e^{-t}} \, dt.

This expression is equivalent to the relation

\Gamma(s) \zeta(s, a) = \int_0^\infty t^{s-1} \frac{e^{-a t}}{1 - e^{-t}} \, dt,

valid under the same conditions, and provides a means to extend the function analytically by deforming contours or using properties of the gamma function. For the full meromorphic continuation to the complex s-plane (with a simple pole at s=1), a Hankel contour integral representation is employed. The contour H starts at +\infty along the upper side of the positive real axis, encircles the origin counterclockwise in a small circle, and returns to +\infty along the lower side. For all complex s \neq 1,

\zeta(s, a) = \frac{\Gamma(1-s)}{2\pi i} \int_H (-z)^{s-1} \frac{e^{-a z}}{1 - e^{-z}} \, dz,

where the branch of (-z)^{s-1} is defined with \arg(-z) \in (-\pi, \pi). This form captures the residues at the poles of \Gamma(1-s) corresponding to the trivial zeros of the zeta function and is particularly useful for asymptotic analysis and numerical computation in the critical strip. Additional integral representations can be derived using the Poisson summation formula applied to suitable generating functions, yielding expressions that relate the Hurwitz zeta to periodic sums or theta-like integrals. For instance, such methods produce contour integrals over alternative paths that incorporate Fourier transforms of the parameter a, enhancing convergence for specific rational values of a. These forms are instrumental in proving reflection formulas and evaluating special cases without relying on the series expansion.^[11]

Taylor and Laurent Series

The Hurwitz zeta function exhibits a simple pole at s = 1 with residue 1 for any fixed a > 0. The Laurent series expansion around this pole is given by

\zeta(s, a) = \frac{1}{s-1} + \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} \gamma_k(a) (s-1)^k,

where the coefficients \gamma_k(a) are known as the generalized Stieltjes constants. These constants generalize the ordinary Stieltjes constants \gamma_k = \gamma_k(1), which appear in the corresponding expansion for the Riemann zeta function \zeta(s). The series converges for all s in the complex plane except at the pole s = 1.^[12] For fixed s \neq 1 with \operatorname{Re}(s) > 1, the Hurwitz zeta function is analytic in the parameter a and admits a Taylor series expansion around any point a_0 > 0. Differentiating the defining Dirichlet series term by term yields the relation

\frac{\partial}{\partial a} \zeta(s, a) = -s \zeta(s+1, a),

with higher-order derivatives following recursively as

\frac{\partial^m}{\partial a^m} \zeta(s, a) = (-1)^m (s)_m \zeta(s + m, a),

where (s)_m = s(s+1) \cdots (s+m-1) denotes the rising Pochhammer symbol. Consequently, the Taylor expansion in a small increment h around a is

\zeta(s, a + h) = \sum_{m=0}^{\infty} \frac{(-h)^m}{m!} (s)_m \zeta(s + m, a).

This expansion is obtained by term-by-term application of the binomial theorem to the series definition and holds by analytic continuation beyond the initial region of convergence.^[13]^[14] When expanding around a = 1, the formula specializes to

\zeta(s, 1 + h) = \sum_{m=0}^{\infty} \frac{(-h)^m}{m!} (s)_m \zeta(s + m),

relating the Hurwitz zeta directly to values of the Riemann zeta function at shifted arguments. The coefficients in this case connect to derivatives of the Riemann zeta function through integral representations or limits involving logarithms, providing a bridge between the two functions.^[12] For fixed s \neq 1, the Hurwitz zeta function is holomorphic in a for \operatorname{Re}(a) > 0, and admits analytic continuation to the complex a-plane with a branch cut along the non-positive real axis. Uniform convergence holds on compact subsets of the a-plane away from the branch cut, facilitating numerical evaluations and asymptotic analyses.^[2]

Functional Equations

Hurwitz Formula

The Hurwitz formula provides a reflection-type functional equation that relates the values of the Hurwitz zeta function \zeta(s, a) at s and $1 - s for complex s with real part not equal to 1 and $0 < a \leq 1. It states that

\zeta(1 - s, a) = \frac{\Gamma(s)}{(2\pi)^s} \left[ e^{-\pi i s / 2} \zeta(s, a) + e^{\pi i s / 2} \zeta(s, 1 - a) \right].

^[1] This equation generalizes the classical functional equation for the Riemann zeta function, which is recovered when a = 1. An equivalent form expresses the relation using sums involving cosines and sines, reflecting the underlying Fourier analysis.^[1] The derivation of the Hurwitz formula typically proceeds via the Poisson summation formula applied to a Gaussian theta function associated with the lattice shifted by a, or equivalently, through the Fourier series expansion of the periodic extension of the function x^{s-1} over [0, 1). These methods exploit the periodicity and the Mellin transform to connect the Dirichlet series representation to an integral form, yielding the reflection relation after applying the gamma function's properties.^[15] The integral representations of \zeta(s, a) play a supporting role in this process.^[2] This functional equation was first formulated by Adolf Hurwitz in 1882 as part of his investigations into generalizations of the Riemann zeta function.^[2]

Functional Equation for Rational Parameters

When the parameter a in the Hurwitz zeta function \zeta(s, a) is rational, say a = r/q where r and q are positive integers with $1 \leq r \leq q and \gcd(r, q) = 1, the general functional equation simplifies to a closed-form expression involving a finite sum over other Hurwitz zeta values.^[16] Specifically, for \operatorname{Re}(s) > 1 and s \neq 0,

\begin{aligned} \zeta\left(1 - s, \frac{r}{q}\right) &= \frac{2 \Gamma(s)}{(2\pi q)^s} \sum_{k=1}^{q} \cos\left(\frac{\pi s}{2} - \frac{2\pi k r}{q}\right) \zeta\left(s, \frac{k}{q}\right). \end{aligned}

This relation, derived from the Hermite representation of the general Hurwitz formula, expresses the value at $1 - s in terms of a sum of q terms, each a Hurwitz zeta function evaluated at shifted rational arguments. The cosine factors arise from the real part of complex exponential terms, connecting to Gauss sums over the residues modulo q.^[16] For rational shifts, the Hurwitz zeta function itself admits an explicit finite-sum representation as a linear combination of Dirichlet L-functions associated to characters modulo q:

\zeta\left(s, \frac{r}{q}\right) = \frac{q^s}{\phi(q)} \sum_{\chi \pmod{q}} \overline{\chi}(r) L(s, \chi),

where the sum runs over all Dirichlet characters \chi modulo q, \phi is Euler's totient function, and L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s}. This decomposition links the functional equation directly to properties of L-functions, facilitating analysis via character sums. A notable example occurs for q = 2 and r = 1, so a = 1/2:

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

where \zeta(s) is the Riemann zeta function. This follows from separating the series for \zeta(s) into terms over even and odd integers, with the even terms contributing $2^{-s} \zeta(s) and the odd terms yielding the factor $2^s - 1.^[16] Substituting into the functional equation provides explicit evaluations, such as relating \zeta(1 - s, 1/2) to \zeta(s). These finite-sum forms enhance computational efficiency, as evaluating \zeta(1 - s, r/q) reduces to computing q Hurwitz zeta values at \operatorname{Re}(s) > 1, where the defining series converges rapidly, or alternatively to sums over L-functions modulo q, avoiding infinite series for the left-hand side.^[16] This utility is particularly valuable in numerical algorithms for analytic continuation in the critical strip.

Particular Values

Values at Negative Integers

The values of the Hurwitz zeta function at negative integers s = -n, where n is a positive integer and $0 < a \leq 1, are given explicitly by the formula

\zeta(-n, a) = -\frac{B_{n+1}(a)}{n+1},

where B_m(x) denotes the m-th Bernoulli polynomial.^[17]^[2] This closed-form expression arises from the analytic continuation of the Hurwitz zeta function and provides rational values depending on a. This formula reveals that \zeta(-n, a) is a polynomial in a of degree n+1. Additionally, since the Bernoulli polynomials satisfy the relation B_{n+1}(a+1) - B_{n+1}(a) = (n+1) a^n, the values of \zeta(-n, a) are periodic with period 1 when a is reduced modulo 1, meaning they depend only on the fractional part of a.^[17] A derivation of this relation can be sketched using the generating function for the Bernoulli polynomials,

\frac{t e^{a t}}{e^t - 1} = \sum_{m=0}^\infty B_m(a) \frac{t^m}{m!},

combined with the integral representation of the Hurwitz zeta function,

\zeta(s, a) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1} e^{-a t}}{1 - e^{-t}} \, dt \quad (\operatorname{Re} s > 1),

and subsequent analytic continuation to s = -n. Alternatively, it follows directly from substituting s = -n into the series expansion derived from the functional equation for the Hurwitz zeta function.^[17]^[2] Representative examples illustrate the formula. For n=1,

\zeta(-1, a) = -\frac{B_2(a)}{2} = -\frac{a^2 - a + \frac{1}{6}}{2},

since B_2(a) = a^2 - a + \frac{1}{6}. For n=0 (corresponding to s=0),

\zeta(0, a) = -B_1(a) = \frac{1}{2} - a,

using B_1(a) = a - \frac{1}{2}. These cases highlight the polynomial nature and provide explicit rational expressions for specific a.^[17]

Derivatives and Rational Values

The derivative of the Hurwitz zeta function with respect to the order s at s = 0 is given by

\zeta'(0, a) = \ln \Gamma(a) - \frac{1}{2} \ln (2\pi),

where \Gamma(a) is the gamma function. This expression follows from the analytic continuation and properties of the function, generalizing the known result for the Riemann zeta function where a = 1.^[1] Near s = 1, the Hurwitz zeta function exhibits a simple pole with residue 1, and its Laurent series expansion is

\zeta(s, a) = \frac{1}{s-1} + \sum_{n=0}^{\infty} \frac{(-1)^n \gamma_n(a)}{n!} (s-1)^n,

where the coefficients \gamma_n(a) are the generalized Stieltjes constants. The leading constant term corresponds to \gamma_0(a) = -\psi(a), with \psi(a) denoting the digamma function, which is the logarithmic derivative of the gamma function. Higher-order terms \gamma_n(a) for n \geq 1 generalize the Stieltjes constants of the Riemann zeta function and relate to generalized harmonic numbers through the expansion's connection to polygamma functions, as \psi^{(n)}(a) = (-1)^{n+1} n! \zeta(n+1, a) for positive integers n. These constants capture the singular behavior and are used in asymptotic analyses.^[18] For rational values of s > 1 with rational parameter a = p/q in lowest terms, the Hurwitz zeta function can often be expressed in terms of the Riemann zeta function, polylogarithms at roots of unity, or multiple zeta values. A key relation is the decomposition via polylogarithms:

\text{Li}_s \left( e^{2\pi i p / q} \right) = q^{-s} \sum_{n=1}^{q} e^{2\pi i n p / q} \zeta(s, n/q),

which allows evaluation of \zeta(s, p/q) using known values of the polylogarithm at primitive q-th roots of unity. For specific cases like s = 2 and a = 1/2, the value simplifies to \zeta(2, 1/2) = (2^2 - 1) \zeta(2) = \pi^2 / 2. Similarly, for a = 1/4, \zeta(2, 1/4) = \pi^2 + 8 [G](/page/Catalan's_constant), where G is Catalan's constant, and Catalan's constant itself satisfies G = [\zeta(2, 1/4) - \zeta(2, 3/4)] / 16. These expressions highlight connections to transcendental constants and are derived from multiplication theorems and Fourier expansions.^[19]^[20] For rational s < 1, such as s = 1/2, closed-form expressions are generally unavailable, but numerical approximations can be obtained via the functional equation or accelerated series, yielding values like \zeta(1/2, 1) \approx -1.46035 and \zeta(1/2, 1/2) \approx -0.824187. These approximations facilitate computational studies and relate to multiple zeta values through generalizations of the reflection formula.^[19]

Relations to Other Functions

Connection to Jacobi Theta Function

The Jacobi theta function \theta_3(z \mid \tau), one of the four classical Jacobi theta functions, is defined as the infinite sum

\theta_3(z \mid \tau) = \sum_{n=-\infty}^{\infty} \exp\left(2\pi i n z + \pi i \tau n^2\right),

where \tau \in \mathbb{C} with \operatorname{Im}(\tau) > 0 and z \in \mathbb{C}. Equivalently, it can be expressed in terms of the nome q = \exp(\pi i \tau) as \theta_3(z, q) = \sum_{n=-\infty}^{\infty} q^{n^2} e^{2\pi i n z}. This function arises naturally in the theory of elliptic functions and modular forms, and its transformation properties under the modular group SL(2, \mathbb{Z}) play a key role in connecting it to zeta functions.^[21] A fundamental link between the Hurwitz zeta function \zeta(s, a) and the Jacobi theta function is provided by an integral representation obtained via the Mellin transform. Specifically, the even variant of the Hurwitz zeta function, defined as

\zeta_{\mathrm{ev}}(s, a) = \frac{1}{2} \sum_{\substack{n \in \mathbb{Z} \\ n + a \neq 0}} \frac{1}{|n + a|^s}

for \operatorname{Re}(s) > 1 and $0 < a \leq 1, admits the representation

\pi^{-s/2} \Gamma\left(\frac{s}{2}\right) \zeta_{\mathrm{ev}}(s, a) = \int_0^\infty t^{s/2 - 1} \frac{\theta_3(a, e^{-\pi t}) - 1}{2} \, dt.

This formula generalizes the well-known integral for the Riemann zeta function (the case a = 1) and allows analytic continuation of \zeta(s, a) to the entire complex plane except for a simple pole at s = 1. An odd variant \zeta_{\mathrm{odd}}(s, a) can be similarly expressed using the derivative \theta_3'(0 \mid \tau).^[21] The derivation of this representation relies on Poisson summation and the Mellin transform. Consider the Gaussian function g(x) = \exp(-\pi x^2 t) for t > 0; its Fourier transform is \hat{g}(\xi) = t^{-1/2} \exp(-\pi \xi^2 / t), leading via Poisson summation \sum_{n \in \mathbb{Z}} g(n + a) = t^{-1/2} \sum_{m \in \mathbb{Z}} \hat{g}(m) e^{2\pi i m a} to an expression involving the Jacobi theta function \theta_3(a, e^{-\pi / t}). Applying the Mellin transform to both sides yields the integral relation for \zeta_{\mathrm{ev}}(s, a), with the subtraction of the n=0 term ensuring convergence. This approach highlights the modular transformation law of the theta function, \theta_3(z \mid -1/\tau) = (-i \tau)^{1/2} e^{\pi i z^2 / \tau} \theta_3(z/\tau \mid \tau), which underpins the functional equation of the Hurwitz zeta.^[21] For rational parameters a = p/q with p, q \in \mathbb{Z}, $1 \leq p < q, and \gcd(p, q) = 1, the connection to the Jacobi theta function facilitates the use of modular properties to derive explicit functional equations. Summing over shifted arguments \theta_3(k/q, e^{-\pi t}) for k = 1, \dots, q exploits the transformation laws, yielding relations that express \zeta(1 - s, a) in terms of \zeta(s, k/q) weighted by trigonometric factors, as in the Hurwitz formula. These properties are essential for evaluating particular values and understanding the distribution of zeros in the context of modular forms.^[21]

Connection to Dirichlet L-Functions

The Hurwitz zeta function \zeta(s, a) provides a generalization of the Riemann zeta function and establishes a direct connection to Dirichlet L-functions through Fourier analysis over Dirichlet characters when a is rational. For a Dirichlet character \chi modulo q, the L-function L(s, \chi) can be expressed as a finite linear combination of Hurwitz zeta functions evaluated at rational arguments:

L(s, \chi) = q^{-s} \sum_{k=1}^q \chi(k) \, \zeta\left(s, \frac{k}{q}\right).

This relation holds for \Re(s) > 1 by the defining series and extends meromorphically to the complex plane using the analytic continuation of \zeta(s, a). The orthogonality relations among Dirichlet characters modulo q allow inversion of this formula, expressing the Hurwitz zeta function in terms of L-functions. Specifically,

\zeta\left(s, \frac{k}{q}\right) = \frac{q^s}{\varphi(q)} \sum_{\chi \bmod q} \overline{\chi}(k) \, L(s, \chi),

where the sum runs over all Dirichlet characters \chi modulo q and \varphi denotes Euler's totient function. This decomposition holds for $1 \leq k \leq q and \Re(s) > 1, with meromorphic continuation following similarly. For primitive characters, the sum restricts to the primitive characters modulo q, adjusted by the conductor. Adolf Hurwitz introduced the zeta function in 1882, motivated by the need to analytically continue and derive functional equations for Dirichlet L-functions at rational parameters, thereby facilitating deeper study of their properties. These relations underpin historical developments in analytic number theory, including Dirichlet's class number formula for imaginary quadratic fields, where L(1, \chi) for the non-principal character \chi modulo the discriminant determines the class number up to explicit factors.

Zeros and Distribution

Zeros

The Hurwitz zeta function \zeta(s, a) exhibits trivial zeros on the negative real axis. For the special case a = 1, where it reduces to the Riemann zeta function, these trivial zeros occur precisely at the negative even integers s = -2m for integers m \geq 1, and they are simple. These locations arise from the functional equation and the properties of Bernoulli numbers. For general $0 < a < 1 with a \neq 1/2, there is exactly one simple real zero in each interval [-2m-2, -2m) for m = 0, 1, 2, \dots.^[22]^[23] The non-trivial zeros of \zeta(s, a) lie in the critical strip where $0 < \operatorname{Re}(s) < 1. For fixed a, the functional equation (available explicitly when a is rational) implies a symmetry in the distribution of these zeros with respect to the line \operatorname{Re}(s) = 1/2. When a = 1, these coincide with the non-trivial zeros of the Riemann zeta function. For general a, the zeros exhibit a similar asymptotic distribution to those of the Riemann zeta function, with the density of zeros up to imaginary part T approximately \frac{\log T}{2\pi}.^[24]

Riemann Hypothesis Analogues

The analogue of the Riemann hypothesis for the Hurwitz zeta function posits that all non-trivial zeros lie on the critical line \operatorname{Re}(s) = 1/2. This statement is false when the parameter a is not an integer. For rational a \neq 1/2, 1, Davenport and Heilbronn established that there are infinitely many zeros with \operatorname{Re}(s) > 1/2.^[25] The result extends to irrational algebraic a by Cassels and to transcendental a by Davenport and Heilbronn.^[26]^[25] Quantitatively, the number of such zeros in the strip $1 < \operatorname{Re}(s) < 1 + \delta up to height T is asymptotically \sim T for fixed \delta > 0.^[27] When a is rational, say a = p/q in lowest terms, the Hurwitz zeta function decomposes as a finite linear combination of Dirichlet L-functions associated with the characters modulo q: specifically, \zeta(s, a) = q^{-s} \sum_{\chi \pmod{q}} \overline{\chi}(p) L(s, \chi), where the sum runs over Dirichlet characters \chi.^[28] The non-trivial zeros of \zeta(s, a) thus occur where this combination vanishes, and their distribution is intimately tied to the zeros of the constituent L-functions. The generalized Riemann hypothesis for these L-functions—that all their non-trivial zeros lie on \operatorname{Re}(s) = 1/2—would imply bounds on the growth and moments of \zeta(s, a), but does not preclude off-critical-line zeros for the Hurwitz zeta itself due to possible cancellations in the sum. Conversely, the known off-line zeros of \zeta(s, a) provide evidence against a direct equivalence, though they inform zero-detection methods for the L-functions.^[28] Unconditional zero-free regions exist near the line \operatorname{Re}(s) = 1. This region widens as |t| increases and underpins prime number theorems in arithmetic progressions via connections to L-functions. Additionally, \zeta(s, a) is zero-free for \operatorname{Re}(s) \geq 1 + a.^[29] Partial results on the density of zeros on the critical line have advanced understanding of their distribution. For specific rational a such as $1/3, 2/3, 1/4, 3/4, 1/6, 5/6, Gonek showed that the number of zeros on \operatorname{Re}(s) = 1/2 up to height T is asymptotically (c + o(1)) \frac{T}{\log T} with $0 < c < 1, confirming a positive but sub-maximal density.^[27] A related conjecture posits that, for rational a \neq 1/2, this number is o(T), assuming no shared zeros among inequivalent L-functions in the decomposition. Recent work has explored zero densities for related Epstein zeta functions (sums of Hurwitz zetas over quadratic forms), yielding improved estimates in strips near \operatorname{Re}(s) = 1/2. For instance, Gonek and Lee established zero-density bounds N(\sigma, T) \ll T^{A(1-\sigma) + \epsilon} for \sigma > 1/2, with explicit constants A depending on the quadratic form. These inform analogous bounds for Hurwitz zetas with quadratic rational parameters.^[30]

Applications

Finite Sums and Identities

One fundamental identity for the Hurwitz zeta function, valid for positive integers a < b and \operatorname{Re}(s) > 1, is the difference formula

\zeta(s, a) - \zeta(s, b) = \sum_{k=a}^{b-1} k^{-s}.

This relation follows directly from the defining Dirichlet series \zeta(s, a) = \sum_{n=0}^\infty (n + a)^{-s}, as the terms telescope when subtracting the series for \zeta(s, b).^[2] Partial sum representations connect the Hurwitz zeta function to generalized harmonic numbers H_m^{(s)} = \sum_{k=1}^m k^{-s}. For positive integer a \geq 1 and positive integer N,

\zeta(s, a) = H_{N+a-1}^{(s)} + \zeta(s, N + a),

with \operatorname{Re}(s) > 1. This expresses the full series as a finite sum plus a tail, facilitating numerical evaluation by approximating the remainder \zeta(s, N + a). For large N, the tail admits the leading approximation

\zeta(s, N + a) \approx \frac{(N + a - 1)^{1 - s}}{s - 1},

derived from the integral tail \int_{N+a-1}^\infty x^{-s} \, dx, with higher-order corrections available via the Euler-Maclaurin formula.^[31] Identities involving sums of Hurwitz zeta functions at rational arguments provide relations to the Riemann zeta function. Specifically, for positive integer n and \operatorname{Re}(s) > 1,

\sum_{k=1}^n \zeta\left(s, \frac{k}{n}\right) = n^s \zeta(s).

This equality arises by reindexing the Dirichlet series for \zeta(s), grouping terms according to residues modulo n, and scaling by the factor n^s from the argument shift.^[32]

Discrete Fourier Transform

The Hurwitz zeta function at rational arguments a = k/N, where k = 1, 2, \dots, N-1 and N is a positive integer, admits an efficient numerical representation via the discrete Fourier transform, leveraging the periodicity of the exponential terms in the defining series. This approach exploits the finite period N to transform the infinite series into a form amenable to fast computation. The key formula arises from applying the discrete Fourier transform to the series expansion. Consider the partial sum approximation \zeta(s, k/N; M) = \sum_{n=0}^{M-1} (n + k/N)^{-s}, but to accelerate, interchange the sums after scaling:

\sum_{k=0}^{N-1} e^{2\pi i m k / N} \zeta(s, k/N) = N^{s} \sum_{l=1}^{\infty} l^{-s} e^{2\pi i m l / N},

where the left side is the DFT of the sequence \zeta(s, k/N), and the right side is a twisted Riemann zeta sum G(s, m/N) = \sum_{l=1}^{\infty} l^{-s} e^{2\pi i m l / N}. Inverting the DFT yields the representation

\zeta\left(s, \frac{k}{N}\right) = \frac{N^{-s}}{N} \sum_{m=0}^{N-1} e^{-2\pi i m k / N} G\left(s, \frac{m}{N}\right).

This adjusted form accounts for the full period by including m = 0, where G(s, 0) = \zeta(s) reduces directly to the Riemann zeta function, with the remaining terms being analogous twisted sums that converge similarly for \operatorname{Re}(s) > 1. For $0 < k < N, the fractional part \{k/N\} = k/N is implicit in the argument.^[33] For numerical evaluation, the algorithm proceeds by approximating the infinite twisted sums G(s, m/N) with partial sums up to a truncation M \gg N, where the partial G_M(s, m/N) = \sum_{l=1}^{M} l^{-s} e^{2\pi i m l / N} is the discrete Fourier transform of the finite sequence l^{-s} for l = 1 to M (padded appropriately). This DFT can be computed using the fast Fourier transform (FFT) in O(M \log M) operations. The tail beyond M is estimated using the Euler-Maclaurin formula or integral remainder for high accuracy. The inverse DFT then yields all \zeta(s, k/N) for k = 0 to N-1 in O(N \log N) operations. For a single value, the full set is still computed, but the method excels when multiple values modulo N are needed, such as in evaluating Dirichlet L-functions via L(s, \chi) = N^{-s} \sum_{k=1}^{N} \overline{\chi}(k) \zeta(s, k/N). The overall complexity for high-precision computation (precision P bits, requiring M \sim 2^P) is dominated by O(M \log M) arithmetic operations, achieving near-optimal scaling.^[34] This FFT-based approach was developed in the 1990s for efficient computation of the Riemann zeta function and its twists, notably in algorithms for locating zeros and multiple evaluations. It was extended to the Hurwitz zeta function in the 2000s for broader applications in analytic number theory, including high-precision libraries like FLINT and Arb, where the periodicity enables reduced computational overhead compared to direct series summation.^[33]

Other Applications

In number theory, the Hurwitz zeta function plays a key role in explicit class number formulas for imaginary quadratic fields through its relation to Dirichlet L-functions. Specifically, for a fundamental discriminant D < 0, the class number h(D) of the imaginary quadratic field \mathbb{Q}(\sqrt{D}) is given by h(D) = \frac{w \sqrt{|D|}}{2\pi} L(1, \chi_D), where w is the number of units and \chi_D is the Kronecker character associated to D.^[35] The L-function L(s, \chi_D) can be expressed as a finite linear combination of shifted Hurwitz zeta functions: L(s, \chi) = q^{-s} \sum_{k=1}^q \overline{\chi}(k) \zeta(s, k/q) for a primitive character \chi modulo q, allowing evaluation of L(1, \chi_D) via known values or approximations of the Hurwitz zeta at rational arguments.^[36] In quantum field theory, the Hurwitz zeta function arises in the regularization of spectral sums for the Casimir energy, particularly for fields on finite intervals with shifted boundary conditions. For instance, the Casimir energy for a scalar field on an interval of length L with periodic or Dirichlet conditions parameterized by a shift a involves the analytic continuation of sums like \sum_{n=0}^\infty (n + a)^2, which is related to derivatives of \zeta(-1, a); more directly, for higher-dimensional or electromagnetic cases, expressions such as \zeta(-3, a) appear in the regularization of mode sums over shifted spectra, yielding finite vacuum energies that depend on the parameter a.^[37] This generalization beyond the Riemann zeta function \zeta(-3) accounts for asymmetries or offsets in the geometry, as seen in computations for \phi^4 theories where integrals of Hurwitz zeta products provide exact Casimir forces.^[3] In statistics, the Hurwitz zeta function connects to generalized harmonic means through its role in defining shifted harmonic numbers. The generalized harmonic number of order s starting at a is H_{n}^{(s)}(a) = \zeta(s, a) - \zeta(s, a + n + 1), which for s = -1 relates to cumulative sums used in generalized means, such as the power mean M_p(a, n) = \left( \frac{1}{n} \sum_{k=0}^{n-1} (k + a)^p \right)^{1/p}, whose limiting behavior as n \to \infty involves \zeta(-p, a).^[38] Additionally, in the coupon collector's problem, the Hurwitz zeta emerges in analyzing the maximum waiting time among geometrically distributed variables with rates $1/n, \dots, 1/m; the expected maximum satisfies asymptotic relations derived from the tail probabilities, expressible via \zeta(1 + it, a) for complex shifts, providing precise bounds on collection times in non-uniform settings.^[39] Recent applications in random matrix theory, particularly from 2024 onward, explore the Hurwitz zeta's zero distribution through moment conjectures analogous to those for the Riemann zeta. For irrational shifts \alpha, the fourth moment \int_T^{2T} |\zeta(1/2 + it, \alpha)|^4 dt \sim c(\alpha) T (\log T)^3 aligns with random matrix predictions for unitary ensembles, where zero spacings exhibit GUE statistics; this extends earlier RMT heuristics to shifted zeros, with subconvexity bounds confirming the leading term and suggesting repulsion patterns similar to eigenvalue spacings in non-Hermitian matrices.^[40] In combinatorics, the Hurwitz zeta function facilitates enumeration of lattice points and shifted partitions via generating functions and asymptotic counts. Similarly, for partitions into parts congruent to a \pmod{q}, the generating function \sum p(n; a, q) x^n = \prod_{k \equiv a \pmod{q}} (1 - x^k)^{-1} evaluates at roots of unity to yield q^{-s} \sum_{k=1}^q e^{-2\pi i k a / q} \zeta(s, k/q), enabling exact counts or asymptotics for restricted partition functions with shifts.^[41]

Special Cases and Generalizations

Special Cases

The Hurwitz zeta function \zeta(s, a) reduces to the Riemann zeta function \zeta(s) when a = 1, as \zeta(s, 1) = \sum_{n=1}^\infty n^{-s}.^[1] A notable particular value in this case is \zeta(2, 1) = \zeta(2) = \pi^2 / 6, solving the Basel problem originally posed by Pietro Mengoli and resolved by Leonhard Euler in 1734. The Dirichlet eta function, or alternating zeta function, \eta(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{-s}, arises as a special case of the Hurwitz zeta with a = 1 via the relation \eta(s) = (1 - 2^{1-s}) \zeta(s, 1).^[42] For a = 1/2, the Hurwitz zeta satisfies \zeta(s, 1/2) = (2^s - 1) \zeta(s), connecting it to alternating series through the Dirichlet beta function \beta(s) = \sum_{k=0}^\infty (-1)^k (2k+1)^{-s} = 2^{-s} \eta(s, 1/2), where \eta(s, 1/2) is the alternating Hurwitz zeta.^[1] The Hurwitz zeta function is recovered as a special case of the Lerch transcendent \Phi(z, s, a) = \sum_{n=0}^\infty z^n (n+a)^{-s} by setting z = 1.^[43] This connection highlights the Hurwitz zeta's role in broader summations involving exponential weights. Higher-order polygamma functions \psi^{(m)}(a), defined as the (m+1)-th derivative of \log \Gamma(a), relate directly to the Hurwitz zeta for positive integers m \geq 1 and \operatorname{Re}(a) > 0 via \psi^{(m)}(a) = (-1)^{m+1} m! \zeta(m+1, a).^[44] Generalized Bose-Einstein integrals, which extend the standard forms used in quantum statistics g_\nu(z) = \frac{1}{\Gamma(\nu)} \int_0^\infty \frac{t^{\nu-1}}{z^{-1} e^t - 1} \, dt = \sum_{k=1}^\infty \frac{z^k}{k^\nu}, incorporate the Hurwitz zeta as a special case through their representation in terms of the Hurwitz-Lerch zeta function when the fugacity parameter aligns with z = 1.

Generalizations

The multiple Hurwitz zeta function extends the standard Hurwitz zeta to multiple variables, defined as \zeta(s_1, \dots, s_k; a) = \sum_{n_1 > \dots > n_k \geq 0} \frac{1}{(n_1 + a)^{s_1} \cdots (n_k + a)^{s_k}} for \operatorname{Re}(s_i) > 1 and a > 0, where the sum is over strictly decreasing non-negative integers.^[45]^[46] This function admits a meromorphic continuation to \mathbb{C}^k with poles along the hyperplanes where any s_i = 1.^[46] It generalizes multiple zeta values when a = 1 and plays a role in evaluating special values at non-positive integers through relations to Bernoulli polynomials.^[47] The Barnes multiple zeta function generalizes the Hurwitz zeta by coupling the indices linearly, defined as \zeta_B(s; a, \omega_1, \dots, \omega_N) = \sum_{m_1=0}^\infty \cdots \sum_{m_N=0}^\infty \left(a + \sum_{i=1}^N \omega_i m_i \right)^{-s} for \operatorname{Re}(s) > N, \operatorname{Re}(a) > 0, and positive \omega_i.^[48] This N-dimensional analogue arises in the theory of multiple gamma functions and satisfies functional equations that recover classical Riemann zeta identities as special cases.^[49] Its analytic continuation features simple poles at each s = 1, \dots, N, and it connects to higher-dimensional polylogarithms.^[50] q-Analogues of the Hurwitz zeta function deform the sum using a parameter q with |q| < 1, such as \zeta(s, a : q) = \sum_{n=0}^\infty q^{s(n+1)} [n + a]_q^{-s}, where _q = \frac{1 - q^x}{1 - q}, which converges for \operatorname{Re}(s) > 0.^[51] Alternative forms employ Jackson q-integrals, providing integral representations that facilitate meromorphic continuation and links to basic hypergeometric series.^[52] These q-deformations preserve reflection formulas analogous to the classical case and appear in q-series identities.^[53] In the p-adic setting, the Hurwitz zeta function has non-archimedean analogues defined via interpolation or Mahler expansions, such as the p-adic Hurwitz-type Euler zeta function \zeta_{p,E}(s, a) = \int_{\mathbb{Z}_p} \langle a + x \rangle^{1-s} \, d\mu_{-1}(a), for x \notin \mathbb{Z}_p, where \mu_{-1} is the fermionic p-adic measure.^[54] This function interpolates Euler polynomials at negative integers and extends to multiple variables, with values at positive integers related to p-adic L-functions.^[55] It satisfies distribution relations and converges p-adically for \operatorname{Re}(s) > 0.^[56] Recent developments in operator theory have introduced quantum generalizations of the Hurwitz zeta function, particularly spectral zeta functions for quantum systems like the Rabi model, defined as \zeta_H(s) = \sum_n \lambda_n^{-s} where \lambda_n are eigenvalues of a Hamiltonian operator H.^[57] These trace over the operator spectrum and yield limits involving Hurwitz zeta values, connecting to ground state energies and functional determinants in quantum field theory.^[58] In 2024–2025 studies, such generalizations have been used to derive asymptotic behaviors for quantum Hamiltonians, highlighting relations to L-functions.^[59]

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