Logarithmic integral function

The logarithmic integral function, denoted \li(x), is a special function in mathematics defined as the Cauchy principal value of the integral \li(x) = \pvint_0^x \frac{\mathrm{d}t}{\ln t} for x > 1. It is intimately connected to the exponential integral function through the relation \li(x) = \Ei(\ln x), where \Ei denotes the exponential integral. In analytic number theory, the logarithmic integral holds particular significance due to its role in the prime number theorem, which asserts that the prime-counting function \pi(x)—the number of prime numbers less than or equal to x—satisfies \pi(x) \sim \li(x) as x \to \infty.^[1] This asymptotic equivalence provides a precise estimate for the distribution of primes, outperforming the cruder approximation \pi(x) \sim x / \ln x by capturing higher-order terms in the error.^[2] Historically, the logarithmic integral was first explored by Carl Friedrich Gauss in the early 19th century as a means to model prime distribution, though he did not publish his findings until a posthumous mention in an 1849 letter.^[1] Bernhard Riemann later incorporated a variant of the function in his seminal 1859 paper on the Riemann zeta function, linking it to the analytic continuation and zeros that underpin modern proofs of the prime number theorem.^[3] Subsequent developments, including explicit formulas by Riemann and von Mangoldt, further highlighted its utility in expressing \pi(x) via sums over the nontrivial zeros of the zeta function.^[1]

Definitions and Variants

Principal Logarithmic Integral

The principal logarithmic integral function, denoted \mathrm{li}(x), is defined for real x > 0, x \neq 1, by the integral

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

where the integrand has a singularity at t = 1.^[4]^[3] To handle the singularity at t = 1, the integral is interpreted in the sense of the Cauchy principal value. For x > 1,

\mathrm{li}(x) = \lim_{\varepsilon \to 0^+} \left( \int_0^{1 - \varepsilon} \frac{\mathrm{d}t}{\ln t} + \int_{1 + \varepsilon}^x \frac{\mathrm{d}t}{\ln t} \right).

This principal value ensures the function is well-defined and continuous except at x = 1, where it diverges to -\infty as x \to 1^- and to +\infty as x \to 1^+.^[4]^[3]^[5] For x > 1, the principal logarithmic integral is related to the exponential integral function \mathrm{Ei}(z), defined as the Cauchy principal value \mathrm{Ei}(z) = -\int_{-z}^\infty \frac{e^{-t}}{t} \mathrm{d}t for \arg z \in (-\pi, \pi), by the identity

\mathrm{li}(x) = \mathrm{Ei}(\ln x).

This connection arises from the substitution t = e^u, transforming the integral appropriately while respecting the principal branch of the logarithm.^[4]^[3] In the interval $0 < x < 1, the integral from 0 to x avoids the singularity at t = 1 and is proper, though the integrand approaches 0 from below as t \to 0^+ since \ln t \to -\infty. Consequently, \mathrm{li}(x) is negative for $0 < x < 1, with \lim_{x \to 0^+} \mathrm{li}(x) = 0. The function is defined on [0, \infty) \setminus \{1\} as a real-valued special function, locally integrable away from the singularity.^[3]^[5]

Offset Logarithmic Integral

The offset logarithmic integral, denoted \mathrm{Li}(x), addresses limitations in the principal logarithmic integral \mathrm{li}(x) by introducing a constant shift tailored for number-theoretic applications. It is defined for x > 1 as \mathrm{Li}(x) = \mathrm{li}(x) - \mathrm{li}(2), where \mathrm{li}(2) \approx 1.04516378.^[3] Equivalently, to handle the singularity at t = 1,

\mathrm{Li}(x) = \lim_{\epsilon \to 0^+} \left[ \int_{0}^{1 - \epsilon} \frac{dt}{\ln t} + \int_{1 + \epsilon}^{x} \frac{dt}{\ln t} \right] - \mathrm{li}(2).

^[3] This offset subtracts the fixed value \mathrm{li}(2) to remove the discontinuity and non-monotonic behavior of \mathrm{li}(x) near x = 1, yielding a smooth, monotonically increasing function ideal for asymptotic estimates in prime distribution studies. This yields \mathrm{Li}(2) = 0, and the function is smooth and monotonically increasing for x \geq 2.^[6] For large x, \mathrm{Li}(x) \approx \frac{x}{\ln x}.^[3]

Representations

Integral Representation

The logarithmic integral function, denoted \mathrm{li}(x) for real x > 1, is fundamentally defined by the Cauchy principal value integral

\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),

which handles the singularity at t = 1 where \ln t = 0. This principal value ensures the integral is well-defined despite the pole, enabling numerical evaluation through standard quadrature methods that bypass the discontinuity.^[3] A useful change of variables transforms this representation into a form involving the exponential integral. Substituting u = \ln t, so t = e^u and \mathrm{dt} = e^u \mathrm{du}, yields

\mathrm{li}(x) = \int_{-\infty}^{\ln x} \frac{e^u}{u} \mathrm{du}

for x > 1, where the integral is again understood in the principal value sense across the origin. This substitution directly links \mathrm{li}(x) to the exponential integral function \mathrm{Ei}(z), defined as

\mathrm{Ei}(z) = \mathrm{PV} \int_{-\infty}^z \frac{e^t}{t} \mathrm{dt}

for real z > 0, via the relation \mathrm{li}(x) = \mathrm{Ei}(\ln x). Integrating the power series expansion of \mathrm{Ei}(z) term by term provides an alternative representation derived from the integral form:

\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 series arises naturally from the integral definition of \mathrm{Ei} and converges for x > 1.^[3] For complex arguments, the logarithmic integral \mathrm{li}(z) admits an analytic continuation via a contour integral

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

where the path of integration avoids the branch cut of \ln t (typically along the negative real axis) and detours around the singularity at t = 1 using a principal value or indentation.^[4] This contour-based definition, consistent with \mathrm{li}(z) = \mathrm{Ei}(\ln z) using the principal branch of the logarithm, extends the function to the complex plane except along the cut from 0 to 1 on the positive real axis.

Series Representation

The series representation of the logarithmic integral function is derived from its integral form by means of the substitution t = e^u, which transforms the principal value integral into the exponential integral \operatorname{Ei}(u). Specifically, for x > 0, x \neq 1, \operatorname{li}(x) = \operatorname{Ei}(\ln x), where \operatorname{Ei}(u) admits the Taylor series expansion around u = 0:

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

with \gamma \approx 0.5772156649 denoting the Euler-Mascheroni constant. This expansion is particularly suited for numerical evaluation near x = 1 (corresponding to small |u|), as the logarithmic singularity at u = 0 is isolated in the \ln |u| term, while the power series captures the regular part. The derivation proceeds by expressing \operatorname{Ei}(u) = \pvint_{-\infty}^u \frac{e^v}{v} \, dv and applying repeated integration by parts to the integrand, yielding the singular terms \gamma + \ln |u| and the convergent power series \sum_{k=1}^\infty \frac{u^k}{k \cdot k!}, which arises from term-by-term integration of the exponential series for e^v.^[7] The radius of convergence of the power series is infinite, but for the principal branch in the complex plane, the representation is valid when |\Im u| < \pi to avoid encircling additional branch points introduced by the periodicity of the logarithm. For small x > 0 (away from the branch cut [0,1]), the exponential substitution yields large negative u, where direct use of the series is inefficient; instead, the leading asymptotic behavior is \operatorname{li}(x) \sim -\frac{x}{\ln (1/x)} as x \to 0^+, though the full exponential form remains the primary tool for local computation near the singularity.^[7]

Properties

Special Values

The principal logarithmic integral function, denoted \mathrm{li}(x), exhibits distinct behavior at specific points due to its integral definition involving the singularity at t=1. As x \to 0^+, \mathrm{li}(x) \to 0, reflecting the integral's convergence from the lower limit where \ln t \to -\infty but the measure is small.^[3] Similarly, \mathrm{li}(x) \to -\infty as x \to 1 from either side, owing to the principal value handling of the pole at t=1, where the contributions from below and above the singularity diverge negatively in the limit.^[3] As x \to \infty, \mathrm{li}(x) \to \infty, consistent with the function's unbounded growth.^[3] Numerical evaluations at key points provide concrete insights into \mathrm{li}(x). For instance, at x = e, the principal value is \mathrm{li}(e) = \int_0^e \frac{dt}{\ln t} \approx 1.895117816, computed via its relation to the exponential integral \mathrm{Ei}(1).^[8] At x = 2, \mathrm{li}(2) \approx 1.045163780, a value central to defining the offset variant \mathrm{Li}(x) = \mathrm{li}(x) - \mathrm{li}(2), which ensures \mathrm{Li}(2) = 0.^[3] A notable connection to the Euler-Mascheroni constant \gamma \approx 0.5772156649 appears in the behavior near the singularity: \lim_{x \to 1} \bigl( \mathrm{li}(x) - \ln |\ln x| \bigr) = \gamma. This limit arises from the small-argument expansion of the related exponential integral, \mathrm{Ei}(z) \sim \gamma + \ln |z| + \sum_{k=1}^\infty \frac{z^k}{k \cdot k!} as z \to 0, with z = \ln x. While exact closed-form expressions exist for \mathrm{li}(x) at transcendental points like x = e through the exponential integral, most evaluations at integer or prime arguments lack simple closed forms and rely on numerical computation or series summation. For small integers and primes p, such as p = 2, 3, 5, values are tabulated in mathematical handbooks and databases for precision up to many decimal places, facilitating applications in analysis.^[3]

Point x	\mathrm{li}(x) (approximate)	Notes
e \approx 2.718	1.895117816	Principal value via \mathrm{Ei}(1).^[8]
2	1.045163780	Basis for offset \mathrm{Li}(x).^[3]

Asymptotic Expansion

The logarithmic integral function \operatorname{li}(x) has the leading asymptotic behavior \operatorname{li}(x) \sim \frac{x}{\ln x} as x \to \infty.^[9] This approximation arises from the dominant contribution in the integral representation and establishes the scale of growth for \operatorname{li}(x), which increases slower than any positive power of x but faster than any logarithmic power.^[9] A refined description is provided by the divergent asymptotic series

\operatorname{li}(x) \sim \sum_{k=0}^\infty \frac{k! \, x}{(\ln x)^{k+1}}

as x \to \infty.^[9] This expansion, obtained via repeated integration by parts starting from the integral representation \operatorname{li}(x) = \operatorname{pv} \int_0^x \frac{dt}{\ln t}, captures higher-order corrections to the leading term.^[9] The series diverges for all finite x > 1, as the factorials k! cause the terms to eventually grow without bound; however, it yields successively better approximations when truncated at the optimal point, typically near the minimal term where subsequent terms begin to increase in magnitude.^[9] An equivalent form avoids the principal value by rewriting \operatorname{li}(x) = \operatorname{li}(2) + \int_2^x \frac{dt}{\ln t} for x > 2, where \operatorname{li}(2) is a constant and the integral has no singularity.^[4] The asymptotic series then applies directly to the integral, with the remainder after truncating at the n-th term bounded by a multiple of the next term's magnitude, ensuring controlled error for large x.^[9] This expansion underpins the prime number theorem, which asserts that the prime-counting function \pi(x) \approx \operatorname{li}(x) as x \to \infty, with relative error o(1) or, equivalently, absolute error o\left( \frac{x}{\ln x} \right).^[10] The connection highlights the role of \operatorname{li}(x) in modeling the distribution of primes, where the asymptotic precision of the series informs bounds on the deviation \pi(x) - \operatorname{li}(x).^[10]

Applications

Prime Number Theorem

The prime number theorem (PNT) states that the prime counting function π(x), which gives the number of primes less than or equal to x, is asymptotically equivalent to the offset logarithmic integral Li(x) as x approaches infinity, denoted π(x) ∼ Li(x).^[1] This equivalence implies that the density of primes near x is approximately 1/ln x, providing a precise measure of how primes become sparser among the integers. The theorem was independently proved in 1896 by Jacques Hadamard and Charles Jean de la Vallée Poussin, building on complex analysis techniques involving the Riemann zeta function.^[11] Their proofs established not only the asymptotic relation but also an explicit error bound: π(x) = Li(x) + O(x \exp(-c \sqrt{\ln x})) for some constant c > 0.^[1] The connection between primes and the logarithmic integral predates the formal proof by over a century. In 1792–1793, Carl Friedrich Gauss, at the age of 15 or 16, empirically observed through extensive computations that π(x) ≈ \int_2^x \frac{dt}{\ln t}, which is precisely Li(x), after examining tables of primes and logarithms.^[1] This insight was later reinforced in Bernhard Riemann's 1859 manuscript on the zeta function, where he suggested a more refined explicit formula linking π(x) to Li(x) and the non-trivial zeros of the zeta function ζ(s), though Riemann did not prove the asymptotic.^[12] The 1896 proofs by Hadamard and de la Vallée Poussin confirmed Riemann's conjecture analytically, showing that the zeta function has no zeros on the line Re(s) = 1, which ensures the validity of the prime distribution estimate.^[11] In practice, the offset logarithmic integral Li(x) = li(x) - li(2), where li(x) is the principal value, provides a superior approximation to π(x) compared to li(x), as it avoids the extraneous contribution from 0 to 2 and aligns better with the cumulative prime count starting from 2.^[3] The difference π(x) - Li(x) exhibits oscillatory behavior due to the zeros of the zeta function but remains bounded by the error terms from the PNT.^[1] Under the Riemann hypothesis, which posits that all non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2, the error term improves dramatically to O(\sqrt{x} \ln x), offering a much tighter bound on the deviation between π(x) and Li(x).^[13] This conditional refinement underscores the logarithmic integral's pivotal role in probing deeper questions about prime distribution.

Other Number-Theoretic Uses

The difference li(x + y) - li(x) \approx \frac{y}{\ln x} provides a heuristic estimate for the number of primes in short intervals [x, x + y] when y = o(x), which underpins probabilistic models for prime gaps. This approximation suggests that maximal prime gaps up to x are on the order of (\ln x)^2, as conjectured by Cramér in 1936 based on a random model where primes occur with density $1/\ln x. Under this model, the probability of a gap larger than c (\ln x)^2 around x becomes exponentially small for sufficiently large c, influencing subsequent refinements like Granville's adjustment incorporating correlations from the Riemann zeta function. In the context of primes in arithmetic progressions, a variant of the logarithmic integral, defined as li(x; q, a) = \int_2^x \frac{dt}{\phi(q) \ln t} for coprime integers q and a, approximates the distribution guaranteed by Dirichlet's theorem on primes in arithmetic progressions. Specifically, the prime counting function \pi(x; q, a) satisfies \pi(x; q, a) \sim \frac{1}{\phi(q)} li(x), with error terms controlled under assumptions like the generalized Riemann hypothesis. This weighted form adjusts the standard logarithmic integral by the Euler totient function \phi(q) to account for the reduced density in the progression modulo q, enabling asymptotic estimates for the least prime in such progressions via Linnik's theorem and its extensions. The offset logarithmic integral Li(x) = li(x) - li(2), plays a key role in analyzing sign changes of \pi(x) - Li(x). Littlewood proved in 1914 that \pi(x) - Li(x) changes sign infinitely often, implying regions where \pi(x) > Li(x). Skewes established in 1933 an explicit upper bound for the first such crossing point where \pi(x) > Li(x), initially estimated below $10^{10^{10^{34}}}, later refined by te Riele in 1987 to below $6.69 \times 10^{370}, with modern estimates placing it around $1.39 \times 10^{316} as of 2025.^[14] These bounds highlight the oscillatory behavior driven by the non-trivial zeros of the Riemann zeta function, with modern computations confirming the first sign change occurs near $1.39 \times 10^{316}. Integrals involving the logarithmic integral relate to the vertical distribution of the non-trivial zeros of the Riemann zeta function through explicit formulas and Fourier analysis. For instance, logarithmic Fourier integrals of \log |\zeta(1/2 + it)| connect the spacing and clustering of zeros to the prime counting discrepancies modeled by li(x). Such relations underpin pair correlation conjectures, like Montgomery's, which predict zero spacings influencing the error term in the prime number theorem.^[15] Generalizations of the logarithmic integral to weighted forms appear in the study of higher moments of L-functions, where integrals like \int_0^T |L(1/2 + it)|^k \, dt for Dirichlet or automorphic L-functions incorporate logarithmic weights to capture average behavior over families. These weighted moments, heuristically linked to products of logarithmic integrals adjusted for conductor or level, provide insights into subconvexity bounds and the distribution of central values, extending the role of li(x) in prime-related asymptotics to broader analytic number theory contexts.