Prime zeta function

The prime zeta function, denoted P(s), is a Dirichlet series in the complex variable s defined by P(s) = \sum_p p^{-s}, where the sum runs over all prime numbers p. This function generalizes the Riemann zeta function \zeta(s) = \sum_{n=1}^\infty n^{-s} by restricting the summation to prime indices alone, and it arises as the Dirichlet generating function for the characteristic function of the primes.^[1]^[2] Introduced formally by Carl-Erik Fröberg in 1968, the prime zeta function builds on earlier numerical computations, such as those by Merrifield up to 35 terms in 1881 and by Liénard up to 167 terms in 1948.^[1]^[2] It converges absolutely for \Re(s) > 1 and admits an analytic continuation to the half-plane $0 < \Re(s) \leq 1, though it has natural boundaries at \Re(s) = 0 due to the zeros of the Riemann zeta function.^[1]^[2] Singularities occur at points s = 1/k for positive integers k that are square-free, reflecting the distribution of prime powers.^[1] A key relation connects P(s) to the Riemann zeta function via the Euler product: \ln \zeta(s) = \sum_{k=1}^\infty \frac{P(ks)}{k} for \Re(s) > 1, with the inverse given by P(s) = \sum_{k=1}^\infty \frac{[\mu(k)](/page/Möbius_function)}{k} \ln \zeta(ks), where [\mu](/page/Möbius_function) is the Möbius function.^[1]^[2] Notable values include P(2) \approx 0.452247, P(3) \approx 0.174763, and P(4) \approx 0.076993, which appear in contexts like the Mertens constant B_1 = \gamma + \sum_{p} \left( \ln(1 - 1/p) + 1/p \right) \approx 0.261497, where \gamma is the Euler-Mascheroni constant.^[1] The function's behavior near s = 1 is P(1 + [\epsilon](/page/Epsilon)) = -\ln \epsilon + [C](/page/The_Constant) + O(\epsilon), with constant C \approx -0.315718.^[1] These properties make P(s) valuable in analytic number theory for studying prime distributions and generalizations like almost-prime zeta functions.^[1]^[2]

Fundamentals

Definition

The prime zeta function P(s) is defined as the Dirichlet series

P(s) = \sum_{p} p^{-s},

where the sum runs over all prime numbers p. This representation holds in the half-plane \Re(s) > 1, where the series converges absolutely.^[1] First studied by Glaisher (1891) for integer values of s in his work on sums of inverse powers of primes, and formally introduced as the prime zeta function P(s) by Fröberg (1968), P(s) serves as a generating function specifically for the primes in analytic number theory. It parallels the Riemann zeta function \zeta(s) = \sum_{n=1}^\infty n^{-s}, which sums over all positive integers, but restricts the terms to primes alone, thereby encoding distributional properties of the primes.^[1]

Historical Background

The series underlying the prime zeta function was first systematically studied by J. W. L. Glaisher in his 1891 paper on sums of reciprocals of prime powers, focusing on first powers as a key object in the distribution of primes. Earlier numerical computations include those by Merrifield (1881) with up to 35 terms and Liénard (1948) with up to 167 terms. This work built on earlier insights from Leonhard Euler, who in the 1730s developed the infinite product representation of the Riemann zeta function over primes, implicitly linking prime distributions to logarithmic expansions that isolate sums over primes akin to the prime zeta series. Glaisher's analysis occurred amid growing interest in prime distributions, shortly before the prime number theorem was established in 1896, highlighting the function's relevance to understanding asymptotic behaviors of primes. The function was formally introduced by Carl-Erik Fröberg in 1968. In the early 20th century, G. H. Hardy and J. E. Littlewood advanced the study of such sums within analytic number theory, particularly in their contributions on the Riemann zeta function and prime distributions in the 1910s, where they used expansions involving series over primes to derive estimates for prime-counting functions. Their work in the 1910s and 1920s, including proofs of infinitely many zeros on the critical line, underscored the role of prime contributions in decomposing the zeta function's logarithm, aiding proofs related to the prime number theorem. A key milestone came with Hans von Mangoldt's early 1900s formulations of explicit formulas for the Chebyshev function, which connected prime power sums directly to zeta function zeros and incorporated prime logarithmic terms that align with the prime zeta function's structure, providing a bridge between oscillatory prime behaviors and analytic continuations. Following World War II, computational interest surged in the 1950s with the advent of electronic computers, enabling numerical evaluations of prime sums and zeta-related series, as seen in early machine-assisted tables that facilitated verification of prime distribution conjectures.

Core Properties

Relation to the Riemann Zeta Function

The prime zeta function P(s) is intimately connected to the Riemann zeta function \zeta(s) via the Euler product formula, which expresses \zeta(s) as an infinite product over primes: \zeta(s) = \prod_p (1 - p^{-s})^{-1} for \Re(s) > 1. Taking the natural logarithm yields \log \zeta(s) = -\sum_p \log(1 - p^{-s}). Expanding each logarithm as a geometric series gives \log(1 - p^{-s}) = -\sum_{k=1}^\infty \frac{p^{-ks}}{k}, so \log \zeta(s) = \sum_p \sum_{k=1}^\infty \frac{p^{-ks}}{k} = \sum_{n=1}^\infty \frac{P(ns)}{n} for \Re(s) > 1. This series relation highlights how P(s) encodes the contribution of primes to the logarithmic structure of \zeta(s).^[3]^[4] By Möbius inversion applied to this Dirichlet convolution, the relation inverts to express P(s) in terms of \zeta(s): P(s) = \sum_{n=1}^\infty \frac{\mu(n) \log \zeta(ns)}{n} for \Re(s) > 1, where \mu denotes the Möbius function. The leading term is \log \zeta(s), with subsequent terms \mu(n) \log \zeta(ns)/n for n \geq 2 remaining bounded as s \to 1^+ since \Re(ns) \geq 2 > 1. Thus, P(s) \sim \log \zeta(s) near the boundary of convergence.^[4] This connection implies growth estimates for P(s) as s \to 1^+. Since \zeta(s) has a simple pole at s=1 with residue 1, \zeta(s) \sim 1/(s-1), and therefore \log \zeta(s) \sim \log \frac{1}{s-1}. Consequently, P(s) \sim \log \frac{1}{s-1} as s \to 1^+. This logarithmic divergence reflects the slow accumulation of the prime harmonic series and is a direct consequence of the prime number theorem, which equates the asymptotic \pi(x) \sim x / \log x to the pole structure of \zeta(s) at s=1.^[4]^[5]

Specific Values and Constants

The prime zeta function P(s) evaluates to specific numerical values at positive integer arguments, which have been computed to high precision and cataloged in mathematical databases. For instance, P(2) = \sum_p p^{-2} \approx 0.4522474200410655^[6], P(3) = \sum_p p^{-3} \approx 0.1747626392994435^[7], and P(4) = \sum_p p^{-4} \approx 0.0769931397642468^[8]. These values arise from direct summation over primes and serve as benchmarks for numerical methods in analytic number theory, often computed using accelerated series or relations to the Riemann zeta function for verification. One prominent application of these integer values lies in the expression for Artin's constant C_{\text{Artin}}, which quantifies the conjectured density of primes p for which a fixed nonsquare integer is a primitive root modulo p. Specifically, \ln C_{\text{Artin}} = -\sum_{n=2}^\infty \frac{L_n - 1}{n} P(n), where L_n denotes the nth Lucas number defined by L_1 = 1, L_2 = 3, and L_n = L_{n-1} + L_{n-2} for n \geq 3^[9]. This yields C_{\text{Artin}} \approx 0.3739558136192025 and \ln C_{\text{Artin}} \approx -0.985110783337^[9], highlighting P(n) as a key component in evaluating densities related to primitive roots. The prime zeta function also features in the Meissel–Mertens constant M, which appears in Mertens' second theorem stating that \sum_{p \leq x} \frac{1}{p} \sim \log \log x + M as x \to \infty^[10]. Here, M = \gamma - \sum_{n=2}^\infty \frac{P(n)}{n}, with \gamma \approx 0.5772156649 the Euler–Mascheroni constant, giving M \approx 0.2614972128476428^[10]. This connection underscores the role of P(s) in asymptotic prime sum estimates central to sieve theory and distribution results.

Analytic Continuation and Representations

Analytic Continuation and Singularities

The prime zeta function P(s) = \sum_p p^{-s}, initially defined by the Dirichlet series over primes p for \Re(s) > 1, admits an analytic continuation to the half-plane \Re(s) > 0 except at certain logarithmic singularities via its relation to the Riemann zeta function. Specifically, Möbius inversion of the identity \log \zeta(s) = \sum_{n=1}^\infty \frac{P(ns)}{n} yields the expression P(s) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log \zeta(ns), which facilitates the extension.^[4]^[1] This continuation reveals that P(s) possesses logarithmic singularities located at s = \rho / n, where \rho denotes a non-trivial zero of \zeta(s) and n \geq 1 is a square-free positive integer, as well as at s = 1/n for square-free positive integers n, reflecting the zeros and the pole of \zeta(s). The singularity at s=1 arises from the logarithmic singularity of \log \zeta(s) at the pole of \zeta(s), while the other singularities stem from the zeros of \zeta(s) in the critical strip and the pole at s=1 for higher n. These singularities are dense along the imaginary axis, accumulating at every point on the line \Re(s) = 0 due to the unbounded imaginary parts of the \rho and the harmonic distribution of the $1/n. Consequently, \Re(s) = 0 forms a natural boundary for P(s), preventing any analytic continuation across this line into \Re(s) < 0.^[1] Regarding growth behavior, as s \to 0^+ along the real axis, |P(s)| \sim \log(1/|s|), reflecting the increasing density of nearby singularities. Near s = 1, P(s) has a logarithmic singularity asymptotically P(s) \sim \log \frac{1}{s-1} + C for some constant C \approx -0.315718, mirroring the singularity of \log \zeta(s). These properties underscore the intricate connection between the distribution of primes and the zeros of the zeta function.^[4]^[1]

Integral Representations

One important integral representation of the prime zeta function P(s) for \Re(s) > 1 is given by

P(s) = s \int_1^\infty \frac{\pi(x)}{x^{s+1}} \, dx,

where \pi(x) is the prime-counting function, i.e., the number of primes less than or equal to x. This formula arises from partial summation (or Stieltjes integration) applied to the defining series for P(s) = \sum_p p^{-s}, and it can be used to extend the representation to \Re(s) > 0 via the known asymptotic behavior of \pi(x). This form facilitates connections to the distribution of primes and supports asymptotic expansions. Another useful relation is the integral of P(s):

\int_s^\infty P(t) \, dt = \sum_p \frac{1}{p^s \log p},

where the sum is over all primes p. This formula arises from term-by-term integration of the defining series for P(t) = \sum_p p^{-t}, justified by the monotone convergence theorem due to the positive terms. The right-hand side converges absolutely for \Re(s) > 0, providing a tool for asymptotic analysis and numerical evaluation in this half-plane. For example, at s = 2, the value is approximately 0.50778218.^[11] These integral representations enable efficient numerical computation of P(s) for \Re(s) > 0, where the defining series diverges for \Re(s) \le [1](/page/1). Direct quadrature of the integrals provides stable evaluations, while contour integration techniques—shifting paths to avoid the logarithmic singularities at s = 1, 1/2, 1/3, \dots—enhance precision for complex arguments, as explored in early computational studies of the function.^[12]

Derivatives

The first derivative of the prime zeta function P(s) is obtained by term-by-term differentiation of its defining Dirichlet series within the region of absolute convergence \Re(s) > 1:

P'(s) = -\sum_p \frac{\log p}{p^s},

where the sum is over all primes p. This expression converges absolutely for \Re(s) > 1 and admits an analytic continuation to the half-plane \Re(s) > 0, consistent with the continuation properties of P(s) itself. Numerical evaluation at s=2 yields P'(2) \approx -0.493091109496..., corresponding to the negative of the series \sum_p (\log p)/p^2. Higher-order derivatives of P(s) are derived analogously by repeated differentiation under the sum for \Re(s) > 1:

P^{(k)}(s) = (-1)^k \sum_p \frac{(\log p)^k}{p^s},

for positive integers k. These series also extend analytically to \Re(s) > 0. The higher derivatives facilitate the construction of Taylor series expansions of P(s) centered at points within the domain, such as positive integers greater than 1. For instance, the Taylor expansion around s=2,

P(s) = P(2) + P'(2)(s-2) + \frac{P''(2)}{2!}(s-2)^2 + \cdots,

provides a local approximation useful for numerical evaluation and asymptotic studies near that point. Such expansions, leveraging the derivatives, contribute to moment estimates in related analytic number theory contexts, including those involving the Riemann zeta function via logarithmic relations.

Generalizations

Almost-Prime Zeta Functions

The almost-prime zeta functions extend the prime zeta function P(s) to sums over positive integers with a prescribed number of prime factors, counting multiplicity. Specifically, for each positive integer k, the k-almost-prime zeta function is defined as

P_k(s) = \sum_{\Omega(n)=k} n^{-s},

where \Omega(n) denotes the total number of prime factors of n with multiplicity, and the sum converges absolutely for \operatorname{Re}(s) > 1. When k=1, this recovers the prime zeta function P(s). These functions arise in analytic number theory for studying the distribution of almost primes, which are integers with exactly k prime factors (e.g., semiprimes for k=2). The P_k(s) are interconnected with the Riemann zeta function \zeta(s) through generating functions and explicit relations derived from Euler products. The ordinary generating function is

\sum_{k=0}^{\infty} P_k(s) x^k = \prod_p (1 - x p^{-s})^{-1},

where the product runs over all primes p and P_0(s) = 1 (corresponding to the empty product for n=1). Setting x=1 yields \zeta(s). More direct relations express P_k(s) in terms of powers and multiples of the prime zeta function P(s), using the cycle index of the symmetric group to account for multiplicities in factorizations. For instance, when k=2,

P_2(s) = \frac{1}{2} \left( P(s)^2 + P(2s) \right),

reflecting contributions from products of two distinct primes and squares of primes. Numerical evaluations provide insight into these functions. For example, P_2(2) \approx 0.14076043434, which is the sum of the reciprocals of the squares of all semiprimes.^[13] Like P(s), the functions P_k(s) for k \geq 1 admit meromorphic continuation to the half-plane \operatorname{Re}(s) > 0. This is achieved via their expressions as finite sums of products of shifted prime zeta functions P(ms) for integers m \geq 1, inheriting the analytic structure of P(s). The prime zeta function itself continues via Möbius inversion from the relation \log \zeta(s) = \sum_{m=1}^{\infty} P(ms)/m, leading to logarithmic branch points at s = \rho/m, where \rho are the non-trivial zeros of \zeta(s) and m is a positive integer. Thus, the singularities of P_k(s) are similarly tied to the zeros of \zeta(s), with no poles in \operatorname{Re}(s) > 0 but branch cuts emanating from these points.

Prime Modulo Zeta Functions

The prime modulo zeta functions are defined for a positive integer modulus m and an integer residue a coprime to m as

P_{m,a}(s) = \sum_{\substack{p \ \mathrm{prime} \\ p \equiv a \pmod{m}}} p^{-s},

where the sum runs over all primes congruent to a modulo m. This represents a restriction of the prime zeta function P(s) to primes in a specific arithmetic progression, capturing the distribution of primes within residue classes coprime to the modulus. The series converges absolutely for \Re(s) > 1, following from the convergence of the full prime zeta series and the positive density of such primes by Dirichlet's theorem.^[14] These functions connect to Dirichlet L-functions through character twists. Define the Dirichlet-twisted prime zeta function for a Dirichlet character \chi modulo m by P_{m,\chi}(s) = \sum_p \chi(p) p^{-s}, where the sum is over all primes. By orthogonality of the characters,

P_{m,a}(s) = \frac{1}{\phi(m)} \sum_{\chi \bmod m} \overline{\chi}(a) P_{m,\chi}(s),

with a possible adjustment for primes dividing m if a \leq m. Moreover, the logarithmic derivative of the L-function yields

\log L(s, \chi) = \sum_{n=1}^\infty \frac{P_{m,\chi}(n s)}{n}

for \Re(s) > 1, expressing the twisted prime zeta contributions across multiples.^[14]^[15] The analytic continuation of P_{m,a}(s) follows from that of the L-functions and twisted prime zetas, which inherit meromorphic properties from L(s, \chi); for non-principal \chi, L(s, \chi) is entire, enabling continuation of P_{m,\chi}(s) to \mathbb{C} except at branch points related to s = 1. For instance, with m=4 and a=1, P_{4,1}(s) sums over primes like 5, 13, 17, 29 congruent to 1 modulo 4, with approximate values P_{4,1}(2) \approx 0.053814 and P_{4,1}(3) \approx 0.008755, illustrating the function's role in studying quadratic residue classes.^[14]^[15]

Other Variants

One notable extension of the prime zeta function involves multiple variables, generalizing the sum over single primes to products over distinct ordered primes. The multiple prime zeta function is defined as

\zeta_P(s_1, \dots, s_m) = \sum_{1 \leq n_1 < \cdots < n_m} \frac{1}{p_{n_1}^{s_1} \cdots p_{n_m}^{s_m}},

where p_n denotes the nth prime number and the sum runs over strictly increasing sequences of positive integers.^[16] This construction parallels multiple zeta values but restricts the arguments to prime bases. For the two-variable case, a symmetric variant appears as P(s_1, s_2) = \sum_{p < q} \left( p^{-s_1} q^{-s_2} + p^{-s_2} q^{-s_1} \right), capturing unordered pairs of distinct primes and enabling evaluations of multiple prime zeta values up to higher weights through relations to known constants.^[17] Another variant incorporates Dirichlet characters to twist the sum, yielding expressions of the form \sum_p \chi(p) p^{-s} for a non-principal character \chi. This form extends the standard prime zeta by weighting primes according to their residue classes modulo the conductor of \chi, allowing analysis beyond fixed moduli and relating to the distribution of primes in more general arithmetic settings via connections to the logarithmic derivatives of Dirichlet L-functions.^[18] Such twisted sums converge conditionally for \operatorname{Re}(s) = 1 when \chi is non-principal, reflecting the non-vanishing of L(1, \chi) and enabling estimates for prime counts in progressions.^[19] These variants, particularly the standard and twisted prime zeta functions, play a role in analogies between prime distributions and random matrix theory. On the critical line s = 1/2 + i\tau, the prime zeta function exhibits asymptotic normality with a covariance function approximating \log |\zeta(1 + i\Delta)|, where \Delta is the spacing, mirroring statistical repulsion patterns observed in the zeros of the Riemann zeta function and Gaussian unitary ensemble models from quantum chaos.^[20] This connection highlights emerging post-2000 research linking the oscillatory behavior of prime sums to spectral properties in random matrices.^[20]