Ramanujan–Soldner constant

The Ramanujan–Soldner constant, denoted \mu, is the unique positive real number at which the logarithmic integral function equals zero, specifically the solution to \mathrm{li}(\mu) = 0, where \mathrm{li}(x) is defined as the Cauchy principal value \mathrm{PV} \int_0^x \frac{dt}{\ln t}.^[1] Its approximate numerical value is \mu \approx 1.451369234883381050283968485892.^[2] This constant bears the names of two mathematicians who independently computed approximations of it: the German astronomer and mathematician Johann Georg von Soldner (1776–1833), who calculated \mu \approx 1.4513692 in 1809 while investigating approximations to the prime-counting function \pi(x) using the logarithmic integral, following Lorenzo Mascheroni's earlier approximation of 1.45137 in 1792,^[2] and the Indian mathematician Srinivasa Ramanujan (1887–1920), who derived \mu \approx 1.45136380 in his unpublished notebooks during the early 20th century.^[1] Soldner's computation appeared in his treatise Theorie et tables d'une nouvelle fonction transcendante, where he explored the integral's role in estimating the density of primes, predating the full prime number theorem by nearly a century.^[3] Ramanujan's value, slightly less accurate, was documented in the later editions of his notebooks edited by Bruce C. Berndt, highlighting his intuitive grasp of special functions without formal training in advanced analysis.^[1] In number theory, the Ramanujan–Soldner constant plays a foundational role in the asymptotic analysis of primes, as \mathrm{[li](/page/Li)}(x) provides a superior approximation to \pi(x) compared to x / \ln x, with \mu normalizing the integral to avoid the singularity at t=1.^[1] For x > \mu, \mathrm{[li](/page/Li)}(x) = \int_\mu^x \frac{dt}{\ln t}, ensuring the function's well-defined behavior beyond the pole.^[1] The constant also appears in series expansions and relations to other mathematical constants, such as the Euler–Mascheroni constant \gamma, through identities like \gamma = -\ln(\ln \mu) - \sum_{n=1}^\infty \frac{\ln n}{n \cdot n! \mu^n}.^[4] Its transcendence remains unproven.^[5]

Definition

Logarithmic Integral Function

The logarithmic integral function, denoted \mathrm{li}(x), is defined for x > 1 as the Cauchy principal value of the improper integral

\mathrm{li}(x) = \mathrm{PV} \int_0^x \frac{\mathrm{dt}}{\ln t},

where the principal value handles the singularity at t = 1, where \ln t = 0.^[6]^[7] This integral is improper not only due to the pole at t = 1 but also at the lower limit t = 0, where \ln t \to -\infty and the integrand approaches zero from the negative side in an integrable manner. To evaluate it rigorously, the principal value is computed as

\mathrm{li}(x) = \lim_{\epsilon \to 0^+} \left( \int_0^{1 - \epsilon} \frac{\mathrm{dt}}{\ln t} + \int_{1 + \epsilon}^x \frac{\mathrm{dt}}{\ln t} \right),

ensuring the contributions from either side of the singularity are balanced symmetrically.^[7] For $0 < x < 1, the function reduces to the ordinary improper integral \int_0^x \mathrm{dt}/\ln t without the principal value, as no singularity arises in that interval.^[7] An alternative representation is \mathrm{li}(x) = \mathrm{Ei}(\ln x), where \mathrm{Ei}(z) is the exponential integral function, which further underscores the connection to special functions and aids in computational and analytical treatments.^[6] The singularity at t = 1 renders the integrand undefined there, with $1/\ln t \to +\infty as t \to 1^+ and $1/\ln t \to -\infty as t \to 1^-, but the principal value integral converges due to the symmetric cancellation around the pole.^[6] For large x, the asymptotic behavior of \mathrm{li}(x) is dominated by the leading term \mathrm{li}(x) \sim x / \ln x, with higher-order corrections given by the series \mathrm{li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k}.^[8] This expansion arises from integrating by parts or relating to the asymptotic form of the exponential integral. Despite the singularity in its defining integral, \mathrm{li}(x) is strictly increasing for x > 1: by the fundamental theorem of calculus applied to the principal value definition, the derivative is \mathrm{li}'(x) = 1 / \ln x > 0 for x > 1, since \ln x > 0 in this regime, confirming monotonic growth.^[7]

The Constant as a Zero

The Ramanujan–Soldner constant \mu is defined as the unique positive real number satisfying \mathrm{li}(\mu) = 0, where \mathrm{li}(x) denotes the logarithmic integral function.^[1] This constant is approximately 1.451369.^[1] An equivalent characterization arises from the integral representation of the logarithmic integral for x > 1, where \mu is the unique positive real number such that \mathrm{li}(x) = \int_{\mu}^{x} \frac{dt}{\ln t}.^[1]^[9] The existence and uniqueness of \mu stem from the behavior of \mathrm{li}(x) on the interval (1, \infty). Specifically, \lim_{x \to 1^+} \mathrm{li}(x) = -\infty and \lim_{x \to \infty} \mathrm{li}(x) = +\infty, while \mathrm{li}(x) remains continuous and strictly increasing on this interval, as its derivative \mathrm{li}'(x) = 1 / \ln x > 0 for x > 1. The intermediate value theorem thus guarantees at least one zero in (1, \infty), and the strict monotonicity ensures exactly one such zero.^[7]^[9] In notation, the constant is conventionally denoted by \mu, though it is sometimes represented as c.^[1] This \mu specifically refers to the unique positive real zero of \mathrm{li}(x), distinguishing it from the complex zeros of the logarithmic integral \mathrm{li}(z) in the complex plane.^[7]

History

Soldner's Computation

In 1809, the German mathematician and astronomer Johann Georg von Soldner introduced the logarithmic integral function li(x) in his paper Théorie et tables d'une nouvelle fonction transcendante, where he computed extensive tables of its values to explore its properties and potential applications.^[2] Soldner's work focused on developing efficient methods for evaluating li(x), defined as the Cauchy principal value integral \int_0^x \frac{dt}{\ln t}, which he recognized as a transcendental function useful for approximating cumulative quantities in analysis.^[3] This computation was motivated by contemporary interests in integral calculus and its extensions beyond elementary functions, building on earlier efforts by figures like Euler and Laplace.^[3] Central to Soldner's analysis was the discovery of the unique positive zero of li(x), which he approximated numerically as μ ≈ 1.4513692346 through meticulous integration techniques.^[2] Lacking modern computational tools, Soldner employed recursive approximation formulas to build his tables, such as expanding li(a + x) ≈ li(a) + \frac{x}{\ln a} plus higher-order series terms involving the logarithm of (1 + x/a), allowing him to extend evaluations from smaller to larger arguments with controlled error.^[3] He handled the singularity at t=1 via the principal value, ensuring convergence, and generated values up to li(1280) with high precision for the era, achieving about 10 decimal places for μ.^[3] This estimation marked an early quantitative insight into the function's oscillatory behavior around zero. Soldner's computation occurred in the broader context of early 19th-century efforts to understand the distribution of prime numbers, predating the prime number theorem by nearly a century.^[1] He suggested li(x) as a model for the prime counting function π(x), noting its asymptotic growth resembling the density of primes, an idea later echoed by Gauss in correspondence around 1810.^[3] This work represented a pioneering attempt to quantify prime density through integral approximations, influencing subsequent analysts like Bessel and laying groundwork for analytic number theory, though Soldner himself did not fully pursue the prime applications in depth.^[3]

Ramanujan's Reference

In 1913, Srinivasa Ramanujan included work on the distribution of prime numbers in his letter to G. H. Hardy, in which he proposed an approximation for the prime-counting function \pi(x) using the logarithmic integral function, expressed as \pi(x) \approx \mathrm{li}(x).^[10] In this context, Ramanujan identified the unique positive constant \mu (also denoted c) as the point where \mathrm{li}(\mu) = 0, noting its role in adjusting the integral to better align with the count of primes.^[1] Ramanujan computed an approximation for \mu \approx 1.45136380 in his notebooks.^[1] This value reflects his independent derivation through empirical series expansions related to prime asymptotics.^[1] Ramanujan's approach emphasized elementary methods to establish the significance of \mu in prime number estimates, avoiding reliance on the Riemann hypothesis or complex analysis, though his proofs contained errors due to limited exposure to advanced function theory.^[11] His contribution underscored the constant's utility in refining \pi(x) \approx \mathrm{li}(x) for practical computations.^[1] Following Ramanujan's independent discovery and approximation, the constant gained recognition as the Ramanujan–Soldner constant in modern literature, honoring both mathematicians despite Soldner's historical priority.^[2]

Properties

Numerical Value and Approximations

The Ramanujan–Soldner constant \mu is approximately $1.45136923488338105028396848589202744949303228\dots.^[2] Following the early approximations by Soldner and Ramanujan, computational advancements in the mid-20th century enabled higher precision through numerical integration of the logarithmic integral on electronic computers.^[1] By the 1990s, techniques such as optimized quadrature rules had produced values to 10,000 decimal digits.^[12] This evolved to 75,500 digits by 2001 using iterative root-finding algorithms.^[12] In 2015, implementations in high-precision libraries like X-MPIR achieved 100,000 digits in under five minutes on standard hardware.^[13] A significant milestone came in 2013 with the computation of the first 10 million decimal digits via Mathematica's arbitrary-precision arithmetic.^[14] As of 2025, software such as Mathematica supports on-demand evaluation to thousands or millions of digits, with error bounds scaled to the specified precision in built-in functions for the logarithmic integral and root solving.^[15] Basic approximation methods rely on evaluating the logarithmic integral via its Taylor series expansion around x=1 (equivalently, u=\ln x \approx 0.372 for \mu), given by

\text{li}(e^u) = \gamma + \ln|u| + \sum_{k=1}^\infty \frac{u^k}{k \cdot k!},

where \gamma is the Euler-Mascheroni constant, then applying iterative solvers like Newton's method to find the root where li(x)=0.^[7] Asymptotic expansions of li(x) for moderately large x, such as li(x) \sim x / \ln x + x / (\ln x)^2 + \dots, provide starting points for these iterations, ensuring rapid convergence with controlled error estimates.^[7]

Series Expansions and Relations

The logarithmic integral function possesses a useful series expansion for x > 1,

\mathrm{li}(x) = \gamma + \ln \ln x + \sum_{k=1}^{\infty} \frac{(\ln x)^k}{k \cdot k!},

where \gamma is the Euler-Mascheroni constant. This representation, originally derived by Ramanujan, enables the analytical study and numerical solution of \mathrm{li}(\mu) = 0 to determine the Ramanujan–Soldner constant \mu.^[7] Ramanujan also provided a more rapidly convergent series for the same domain,

\mathrm{li}(x) = \gamma + \ln \ln x + \sqrt{x} \sum_{n=1}^{\infty} \frac{(-1)^{n-1} (\ln x)^n}{n! \, 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1}.

Both expansions highlight the connection between \mathrm{li}(x) and exponential generating functions, facilitating approximations near the zero at \mu.^[7] A key integral relation defines \mu implicitly through the principal value integral \mathrm{li}(x) = \mathrm{PV} \int_0^x \frac{dt}{\ln t}, with \mathrm{li}(\mu) = 0. Consequently, for x > \mu,

\mathrm{li}(x) = \int_{\mu}^{x} \frac{dt}{\ln t}.

This equation positions \mu as the effective lower limit that offsets the integral to yield the standard form of \mathrm{li}(x).^[1] The asymptotic series for \mathrm{li}(x) as x \to \infty incorporates \mu via the offset in the integral representation:

\mathrm{li}(x) \sim \int_{\mu}^{x} \frac{dt}{\ln t} \sim \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k}.

The leading terms establish the scale of \mathrm{li}(x) relative to the prime counting function, with \mu ensuring the integral starts from the zero point.^[7] The existence and uniqueness of \mu > 1 follow from the monotonicity of \mathrm{li}(x), as \mathrm{li}'(x) = 1 / \ln x > 0 for x > 1, combined with the limits \lim_{x \to 1^+} \mathrm{li}(x) = -\infty and \lim_{x \to \infty} \mathrm{li}(x) = \infty, without invoking series expansions.^[7]

Applications

Role in Prime Number Theory

The Ramanujan–Soldner constant, denoted μ, arises in the prime number theorem (PNT), which asserts that the prime-counting function π(x), counting the number of primes less than or equal to x, is asymptotically equivalent to the logarithmic integral li(x), so π(x) ∼ li(x) as x → ∞. The function li(x) is defined as the Cauchy principal value integral li(x) = \lim_{\epsilon \to 0^+} \left( \int_0^{1-\epsilon} \frac{dt}{\ln t} + \int_{1+\epsilon}^x \frac{dt}{\ln t} \right) for x > 1, and this li(x) has a unique positive zero at μ ≈ 1.45136923488, where li(μ) = 0. This zero allows an alternative representation li(x) = \int_μ^x \frac{dt}{\ln t} for x > μ, effectively offsetting the lower limit of integration to avoid the singularity at t = 1 and ensuring the function's asymptotic expansion li(x) ∼ \frac{x}{\ln x} + \frac{x}{(\ln x)^2} + \cdots aligns precisely with the distribution of primes.^[1]^[3] In refined approximations beyond the basic PNT, such as the inclusion of secondary terms to account for primes near smaller powers of x, μ adjusts the constant offset in expressions like π(x) ≈ li(x) - li(\sqrt{x}) + li(x^{1/3}) - \cdots. These corrections, derived from Möbius inversion applied to the Chebyshev function, improve the accuracy of prime estimates by incorporating the zero of li(x) into the baseline constant term, reducing the error in comparisons between π(x) and li(x) for moderate x. For instance, numerical evaluations show that shifting by μ enhances the fit for x up to 10^6, where the difference |π(x) - li(x)| is minimized relative to unadjusted forms like x / \ln x.^[7]^[3] The constant also appears implicitly in the explicit formula of von Mangoldt for the weighted prime-counting function ψ(x) = \sum_{n \leq x} \Lambda(n) = x - \sum_\rho \frac{x^\rho}{\rho} - \ln(2\pi) - \frac{1}{2} \ln(1 - x^{-2}), where \Lambda(n) is the von Mangoldt function, the sum is over non-trivial zeros ρ of the Riemann zeta function, and ψ(x) relates to π(x) via partial summation: π(x) \approx \int_2^x \frac{ψ(t)}{t \ln t} , dt. Since the main term x integrates to li(x) under this transformation, the offset defined by μ ensures the principal value aligns the explicit oscillatory corrections with observed prime distributions, providing a foundational link between analytic number theory and the zeta function's zeros.

Connections to Other Mathematical Constants

The Ramanujan–Soldner constant μ exhibits notable analytic relations with the Euler-Mascheroni constant γ, providing expressions that link these two fundamental constants through series derived from the logarithmic integral function. One such relation is given by the infinite series

\gamma = -\ln(\ln \mu) - \sum_{n=1}^\infty \frac{(\ln \mu)^n}{n \cdot n!},

which arises from the condition li(μ) = 0 and converges to over 30 decimal places with just 20 terms when computed using software like PARI/GP.^[4] A second expression, based on Ramanujan's master theorem and involving alternating terms, is

\gamma = -\ln(\ln \mu) + \sqrt{\mu} \sum_{n=1}^\infty (-1)^n \frac{(\ln \mu)^n}{n! \cdot 2^{n-1} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1}}.

This series achieves even higher precision, up to 37 digits with 20 terms, leveraging the rapid convergence properties inherent to factorial denominators.^[4] These formulas stem from integral representations of li(x) and manipulations using the cosine integral or related functions, offering pathways to express γ solely in terms of μ without direct reliance on harmonic numbers or other traditional definitions of γ.^[4] Such relations are particularly valuable in analytic number theory, where they enable efficient high-precision computations of γ by substituting precomputed values of μ, thereby supporting derivations of asymptotic formulas and bounds involving special functions. Additionally, μ connects to the exponential integral function Ei(z) via the identity li(x) = Ei(\ln x) for x > 1, implying that Ei(\ln μ) = 0 defines the constant implicitly.^[1] This tie allows properties of Ei, such as its series expansions around branch points, to inform studies of μ's behavior. While direct links to the Lambert W function appear in broader series developments for μ, they primarily highlight structural similarities in solving transcendental equations rather than explicit closed forms. Overall, these interconnections facilitate proofs of convergence and numerical stability in representations of li(x), aiding theoretical advancements in the analysis of integrals with logarithmic singularities.