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Euler's constant

Euler's constant, also known as the Euler-Mascheroni constant and denoted by the Greek letter γ, is a fundamental approximately equal to 0.5772156649, defined as the limiting difference between the nth H_n = 1 + \frac{1}{2} + \dots + \frac{1}{n} and the natural logarithm of n as n approaches :
\gamma = \lim_{n \to \infty} (H_n - \ln n). Introduced by the Swiss mathematician in 1734 through his work on harmonic progressions, the constant emerged while studying the asymptotic behavior of the harmonic series, where Euler computed its value to 16 decimal places by 1781. Independently investigated by the Italian mathematician in 1790, who calculated it to 32 decimal places (though with errors beyond 19 digits) and introduced the symbol γ, the constant bears his name in recognition of this contribution. The constant plays a pivotal role across diverse mathematical domains, including , where it appears in the Euler-Maclaurin formula relating sums to integrals; in the theory of the , via the \psi(1) = -\gamma; and in the , such as \zeta'(0) = -\frac{1}{2} \ln(2\pi). It also arises in integral representations, such as \gamma = \int_0^\infty \left( \frac{e^{-t}}{t} - \frac{1}{1 + t} \right) \, dt, and in probabilistic contexts like the . Further analyzed through continued fractions and series expansions, γ's irrationality, like its transcendence, remains an open problem, though it is suspected to be transcendental. Its value can be approximated efficiently using accelerated series, such as \gamma \approx \sum_{k=1}^m \left[ \frac{1}{k} - \ln\left(1 + \frac{1}{k}\right) \right] + \ln\left(m + \frac{1}{2}\right).

Definition

Limit involving harmonic numbers

The nth harmonic number H_n is defined as the partial sum H_n = \sum_{k=1}^n \frac{1}{k}, which represents the sum of the reciprocals of the first n positive integers. As n approaches , the sequence H_n diverges to , but its growth is asymptotically tied to the natural logarithm \ln n, the logarithm with base e \approx 2.71828. In 1734, Leonhard Euler introduced what is now known as Euler's constant \gamma by identifying it as the limiting value that captures the difference between this divergent harmonic sum and the logarithmic growth. Specifically, Euler's constant is defined by the limit \gamma = \lim_{n \to \infty} \left( H_n - \ln n \right), where this limit converges to approximately $0.5772156649. This expression establishes \gamma$ as a fundamental constant arising from the subtle interplay between discrete and continuous logarithmic approximation in the analysis of the harmonic series. The significance of this limit is further illuminated by the asymptotic expansion of the harmonic numbers, which reveals \gamma as the constant term in the series describing the divergence of H_n. Euler observed that H_n \approx \ln n + \gamma + \frac{1}{2n} - \sum_{k=1}^\infty \frac{B_{2k}}{2k \, n^{2k}}, where B_{2k} denote the Bernoulli numbers and the expansion provides increasingly accurate approximations for large n. This formulation underscores \gamma's role in refining the rough estimate H_n \sim \ln n, with higher-order terms like $1/(2n) offering corrections that diminish as n grows.

Integral representations

One prominent integral representation of Euler's constant \gamma arises from regularizing the divergent integral associated with the exponential function, yielding the split form \gamma = \int_0^1 \frac{1 - e^{-t}}{t} \, dt - \int_1^\infty \frac{e^{-t}}{t} \, dt. This expression is equivalent to \gamma = \int_0^1 \frac{1 - e^{-t}}{t} \, dt - E_1(1), where E_1(z) = \int_z^\infty \frac{e^{-u}}{u} \, du denotes the function. Another equivalent representation is the single improper integral \gamma = \int_0^\infty e^{-t} \left( \frac{1}{1 - e^{-t}} - \frac{1}{t} \right) dt. A related form, useful for analytic continuations and connections to the , is \gamma = -\int_0^\infty e^{-t} \ln t \, dt. These integrals originate from the limit definition \gamma = \lim_{n \to \infty} (H_n - \ln n), where H_n is the nth , by approximating the discrete sum and logarithm with continuous . Specifically, H_n = \int_0^\infty e^{-t} \frac{1 - e^{-n t}}{1 - e^{-t}} \, dt and \ln n = \int_0^\infty \frac{e^{-t} - e^{-n t}}{t} \, dt; subtracting these and taking n \to \infty cancels the e^{-n t} terms, leaving the integral for \gamma. The representations converge despite singularities at t = 0, where terms like $1/t appear, because the leading singular contributions cancel within the integrand—for instance, \frac{1}{1 - e^{-t}} - \frac{1}{t} \sim \frac{1}{2} as t \to 0^+ in the second form, and \frac{1 - e^{-t}}{t} \to 1 in the first integral of the split form. At infinity, the exponential decay ensures integrability. This contrasts with the original limit definition, where both H_n and \ln n diverge logarithmically, but their difference remains finite; the integrals embed this cancellation continuously, facilitating proofs of properties and numerical evaluations without discrete truncation errors.

History

Euler's discovery

Leonhard Euler first encountered the constant during his early investigations into the asymptotic behavior of the harmonic series in the late 1720s and early 1730s. His work on divergent series, including the harmonic series, intersected with his efforts to solve the Basel problem—determining the sum of the reciprocals of squares—which he addressed around 1734 through generalizations of series expansions. These studies revealed that the partial sums of the harmonic series diverge like the natural logarithm, prompting Euler to examine the limiting difference between the nth harmonic number H_n and \ln n. In his seminal paper "De progressionibus harmonicis observationes," presented to the St. Petersburg Academy of Sciences on , 1734 (and published in ), Euler formally introduced this limit as converging to a finite positive constant, which he computed numerically using differences of partial sums and logarithmic approximations. He arrived at the approximately 0.57721, accurate to five decimal places, by evaluating the expression for sufficiently large n and noting the stabilizing . Euler also provided estimates for his approximation, confirming the constant's convergence, and expressed it via an involving generalized sums s_k = \sum_{m=1}^\infty \frac{1}{m(m+k)}, such as \gamma = \frac{1}{2}s_2 - \frac{1}{3}s_3 + \frac{1}{4}s_4 - \cdots. In 1781, Euler computed the constant to 16 decimal places.

Naming and later contributions

The Euler-Mascheroni constant received its name in recognition of the independent contributions by Leonhard Euler and Lorenzo Mascheroni to its numerical evaluation and notation. In 1790, Mascheroni published computations of the constant to 32 decimal places in his treatise Adnotationes ad calculum integrale Euleri, where he introduced the symbol γ to denote it, though his values contained errors in the 20th–22nd and 31st–32nd positions. During the , further progress illuminated properties of the constant. The marked accelerated numerical exploration and broader applications. With the rise of electronic computers in the and , researchers computed γ to thousands of decimal places using accelerated series methods, such as those based on the ; by the 1960s, values exceeding 1,000 digits were routine, facilitating deeper study of its decimal behavior. In parallel, the constant appeared in during the 1970s, notably in perturbative schemes where it arises in logarithmic divergences and asymptotic expansions for scattering amplitudes. As of 2025, the of γ remains an despite extensive efforts. delivered a comprehensive survey that year on the family of Euler constants, reviewing historical developments and modern generalizations while underscoring the persistent challenges in proving basic arithmetic properties like .

Properties

Irrationality status

The irrationality of the Euler-Mascheroni constant \gamma remains an open problem as of 2025, with no proof that it is either irrational or transcendental. Additionally, \gamma is not a quadratic irrational, a consequence of Nesterenko's 1996 work on the algebraic independence of values of the Gamma function and related constants. Significant progress has been made in bounding how well \gamma can be approximated by rational numbers, which would be relevant if \gamma is irrational. In 1993, Hata proved that for any integers p and q > 0, \left| \gamma - \frac{p}{q} \right| > \frac{1}{q^{7.503}}. This provides an irrationality measure \mu(\gamma) \leq 7.503 + \epsilon for any \epsilon > 0. Minor improvements followed, including Hata's own refinement to approximately 7.606 in 1994, but no substantial advances have lowered the exponent significantly by 2025. These Diophantine approximation bounds indicate that \gamma, if irrational, cannot be approximated by rationals better than a certain rate, distinguishing it from Liouville numbers or other highly approximable irrationals. The results connect indirectly to the Lindemann-Weierstrass theorem via the role of e^\gamma in the Weierstrass form of the Gamma function, though this has not yielded a direct proof of \gamma's irrationality.

Continued fraction expansion

The continued fraction expansion of the Euler-Mascheroni constant \gamma is \gamma = [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, \dots], where the partial quotients a_n are irregular and appear to increase without bound over time. The initial terms of this expansion were first computed by Philipp Ludwig Seidel in 1877 using manual methods based on high-precision approximations of \gamma. In 1977, Richard P. Brent extended the computation significantly, determining the first 20,000 partial quotients through an efficient algorithm involving high-precision evaluation of \gamma via its integral representations and subsequent application of the to extract the continued fraction terms. The irregular nature of the partial quotients, with occasional large values such as and 40, indicates that \gamma admits good rational approximations from its convergents, as smaller quotients early on yield relatively rapid convergence, while the overall growth rate provides insights into the constant's properties and supports arguments toward its by avoiding patterns consistent with rationality or irrationality. The convergents p_n/q_n of this expansion offer successively better rational approximations to \gamma. For instance, the fifth convergent $11/19 \approx 0.57895 approximates \gamma to about three decimal places, while the sixth convergent $15/26 \approx 0.57692 improves accuracy to four decimal places; the seventh convergent $71/123 \approx 0.57724 achieves five decimal places, and higher-order convergents like $3035/5258 \approx 0.577215 provide precision exceeding six decimal places. These convergents satisfy | \gamma - p_n/q_n | < 1/(q_n q_{n+1}), ensuring their effectiveness for computational purposes.

Numerical computation and digits

The numerical value of Euler's constant γ is approximately 0.57721566490153286060651209008240243104215933593992, with the first 50 digits given by 0.57721566490153286060651209008240243104215933593992. Early high-precision computations relied on the to accelerate the evaluation of the defining limit γ = lim (H_n - ln n), where H_n is the nth , by providing asymptotic expansions that reduce the number of terms needed for convergence. In 1952, J. W. Wrench Jr. used this method to compute γ to 328 decimal places, a significant advance at the time given the computational limitations of mechanical calculators. For higher precision, series representations of γ are preferred, evaluated efficiently using binary splitting, a recursive technique that minimizes intermediate expression swell when summing rational terms. This approach was adapted for γ in algorithms developed by R. P. Brent in 1980, building on his earlier work for multiple-precision arithmetic, enabling computations to millions of digits on early computers. Modern implementations, such as those in the y-cruncher software, combine binary splitting with optimized series like the Brent-McMillan variant to achieve record-breaking precision; as of September 2023, γ has been computed to 1.337 trillion (1,337,000,000,000) decimal places using this method on multi-core hardware. A key challenge in direct evaluation of the limit form is subtractive cancellation between H_n and ln n, which requires n ≈ 10^d for d digits of precision, leading to prohibitive storage and time costs for large d. Accelerated methods like or binary-split series circumvent this by converging quadratically or faster without such cancellation.

Representations and identities

Relation to the gamma function

The \gamma is fundamentally connected to the \Gamma(z) via the \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}, the logarithmic derivative of \Gamma(z). In particular, \gamma = -\psi(1). This relation arises from the Weierstrass infinite product representation of the reciprocal gamma function, \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-z/n}, valid for \Re(z) > 0. Taking the natural logarithm gives \ln \Gamma(z) = -\ln z - \gamma z + \sum_{n=1}^{\infty} \left[ \frac{z}{n} - \ln\left(1 + \frac{z}{n}\right) \right]. Differentiating with respect to z yields the series expression for the , \psi(z) = -\frac{1}{z} - \gamma + \sum_{n=1}^{\infty} \left[ \frac{1}{n} - \frac{1}{n + z} \right]. Evaluating at z = 1 produces \psi(1) = -1 - \gamma + \sum_{n=1}^{\infty} \left[ \frac{1}{n} - \frac{1}{n+1} \right]. The infinite sum telescopes to 1, simplifying to \psi(1) = -\gamma. This connection extends naturally to positive integers, where \psi(n+1) = -\gamma + H_n and H_n = \sum_{k=1}^n \frac{1}{k} is the nth . This formula generalizes \gamma by linking it to the harmonic series through special values of the and provides a means to express harmonic numbers in terms of \gamma and \psi.

Relation to the Riemann zeta function

The \zeta(s) possesses a simple at s=1 with residue 1. Its expansion about this point is given by \zeta(s) = \frac{1}{s-1} + \sum_{k=0}^{\infty} \frac{\gamma_k}{k!} (s-1)^k, where \gamma_0 = \gamma is the Euler-Mascheroni constant and \gamma_k for k \geq 1 are the Stieltjes constants. This expansion identifies \gamma as the constant term in the regular part of \zeta(s) at the pole. Equivalently, \gamma arises as the limit \gamma = \lim_{s \to 1^+} \left( \zeta(s) - \frac{1}{s-1} \right), which regularizes the divergent behavior of \zeta(s) as s approaches 1 from the right. This limit form connects directly to the defining limit of \gamma via harmonic numbers, as the partial sums H_n = \sum_{k=1}^n 1/k approximate \zeta(s) for s \to 1^+ in a manner that isolates the logarithmic divergence and the constant \gamma. The appearance of \gamma in this expansion facilitates the of \zeta(s) from its initial domain \operatorname{Re}(s) > 1 to a on the , with the sole pole at s=1; the explicit form of the ensures the is holomorphic elsewhere. Through the of the , which interlinks \zeta(s) and the \Gamma(s), \gamma = -\Gamma'(1) ties indirectly to derivatives such as \zeta'(0) = -\frac{1}{2} \ln(2\pi). This underscores \gamma's in broader analytic properties of \zeta(s).

Integral expressions

Euler's constant \gamma possesses a variety of integral representations that extend beyond elementary forms, often involving double integrals or generalizations of Frullani's integral theorem. These expressions facilitate advanced manipulations in and provide tools for numerical evaluation and proof of properties. One notable double integral representation, analogous to those for the values at even integers, is attributed to Sondow (2005): \gamma = -\int_0^1 \int_0^1 \frac{1 - x}{1 - xy} \log(xy) \, dx \, dy. This form arises from series expansions and analytic continuations related to the Lerch transcendent, highlighting connections to polylogarithms. A Frullani-type integral yielding \gamma is: \gamma = \int_{-\infty}^\infty \frac{\log(1 + e^{-t})}{e^t t^2 + \pi^2} \, dt. This integral, derived using Fourier analysis and residue calculus, connects \gamma to probabilistic interpretations involving harmonic series. Further parameterized forms emerge in limits of Frullani differences. Variants of this appear in derivations of generalized constants, such as \gamma(a) = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k^a} - \int_1^n x^{-a} \, dx \right) for a=1, with integral analogs. Recent work, including preprints from 2025, explores new functional integrals tied to E-harmonic numbers, building on earlier parameterizations for enhanced computational efficiency.

Series expansions

One fundamental series representation of the Euler-Mascheroni constant \gamma is the telescoping series \gamma = \sum_{k=1}^\infty \left( \frac{1}{k} - \ln\left(1 + \frac{1}{k}\right) \right), which follows directly from expanding the definition \gamma = \lim_{n \to \infty} (H_n - \ln n), where H_n is the nth harmonic number, as the partial sum up to N simplifies to H_N - \ln(N+1). An equivalent form is the rearranged telescoping series \gamma = \sum_{n=1}^\infty \left[ \frac{1}{n} - \ln(n+1) + \ln n \right], where the logarithmic terms telescope, leaving the harmonic contribution in the limit. For accelerated convergence, Vacca's series offers improved efficiency: \gamma = \sum_{n=2}^\infty \frac{(-1)^n}{n \lfloor n/2 \rfloor}, where \lfloor \cdot \rfloor denotes the floor function; this form arises from grouping terms in the harmonic series and achieves quadratic convergence relative to the basic alternating series. The denominator \lfloor n/2 \rfloor approximates n/2 for large n, effectively doubling the decay rate compared to $1/n terms alone. High-precision computations of \gamma leverage binary splitting techniques on accelerated series, such as the representation \gamma = \frac{1}{2} \sum_{k=0}^\infty \frac{(-1)^k}{k+1} \ln \left( \frac{2k+2}{2k+1} \right), which originates from differentiating the or related integral forms and allows recursive evaluation of partial sums with minimal intermediate precision loss. This approach, first applied by Gourdon in to compute over 100 million digits, has been refined in modern variants for even larger expansions, enabling calculations beyond $10^{12} digits by 2025 through optimized parallel implementations.

Applications

In analysis

Euler's constant \gamma plays a pivotal role in asymptotic analysis, particularly through the Euler-Maclaurin summation formula, which relates discrete sums to integrals and provides the asymptotic expansion for the harmonic numbers H_n = \sum_{k=1}^n \frac{1}{k}: H_n = \ln n + \gamma + \frac{1}{2n} - \sum_{k=1}^m \frac{B_{2k}}{2k n^{2k}} + R_{m,n}, where B_{2k} are the Bernoulli numbers and the remainder term R_{m,n} satisfies |R_{m,n}| \leq \frac{|B_{2m+2}|}{(2m+2) n^{2m+2}} for sufficiently large n. This expansion highlights \gamma as the constant term bridging the harmonic series and the natural logarithm, enabling precise approximations in summation problems across analysis. In the context of differential equations, \gamma emerges in the asymptotic behavior of solutions to equations like y' + y \ln x = 0, whose explicit y(x) = C x^{-x} e^x (for x > 0) connects to functions whose large-argument expansions incorporate \gamma, such as the via Stirling's series. Similarly, \gamma appears directly in s; for instance, the Laplace transform of \ln t (for t > 0) is \mathcal{L}\{\ln t\}(s) = -\frac{\gamma + \ln s}{s} for \operatorname{Re} s > 0, reflecting \gamma's role in regularizing logarithmic singularities in transform theory. In , \gamma features prominently in the asymptotic expansions of . The \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)} admits the expansion \psi(z) \sim \ln z - \frac{1}{2z} - \sum_{k=1}^\infty \frac{B_{2k}}{2k z^{2k}} as |z| \to \infty with |\arg z| < \pi, and since \psi(1) = -\gamma, this positions \gamma as the fundamental constant in the function's behavior near positive integers. Likewise, in the Hurwitz zeta function \zeta(s, a) = \sum_{n=0}^\infty (n+a)^{-s} for \operatorname{Re} s > 1 and a > 0, the around s=1 is \zeta(s, a) = \frac{1}{s-1} + \sum_{k=0}^\infty \frac{(-1)^k}{k!} \gamma_k(a) (s-1)^k, where the zeroth Stieltjes constant \gamma_0(a) = -\psi(a) generalizes \gamma = \gamma_0(1), linking \gamma to meromorphic continuations in the . Recent advancements (2024–2025) have introduced new families of regularized functionals generalizing \gamma via integral expressions involving Clausen functions and numbers, providing tools for analyzing in integral equations and asymptotic regularization.

In number theory

Euler's constant \gamma plays a significant role in the asymptotic behavior of the product over primes in Mertens' third theorem. Specifically, the product \prod_{p \leq x} \left(1 - \frac{1}{p}\right) \sim \frac{e^{-\gamma}}{\ln x} as x \to \infty, where the product is taken over all primes p \leq x. This result follows from the and the properties of the near s=1, highlighting \gamma's connection to the distribution of primes. In the theory of integer partitions, \gamma appears in asymptotics for certain partition statistics. For reciprocal supernorm partition statistics, the cumulative sums include \gamma in their leading asymptotic behavior, providing refinements to the growth rates. A notable appearance of \gamma occurs in connection with harmonic numbers evaluated at triangular indices. Let t_n = \frac{n(n+1)}{2} denote the nth . Then, H_{t_n} - \ln t_n \to \gamma as n \to \infty, where H_k is the kth . Ramanujan provided an explicit for this difference, given by H_{t_n} = \ln t_n + \gamma + \sum_{k=1}^\infty \frac{(-1)^{k+1}}{k t_n^k} \left(1 + \frac{1}{2 t_n} + \cdots + \frac{1}{k t_n^{k-1}}\right), which converges rapidly and refines the approximation. This identity underscores \gamma's role in bridging harmonic sums and for quadratic indices. Regarding Diophantine approximations, its remains unproven, and bounds on its rational approximations have been established. For example, there exist infinitely many rationals p/q such that |\gamma - p/q| < 1/(q^2 \ln q), but \gamma satisfies effective irrationality measures excluding Liouville-type approximations. Further, rational approximations to \gamma can be constructed using Hankel determinants of specific sequences, yielding convergents with controlled error terms in the continued fraction expansion. These results provide quantitative insights into the arithmetic nature of \gamma in number-theoretic contexts.

In other mathematical fields

In probability theory, Euler's constant appears in the Erdős–Kac theorem, which describes the normal order of the number of distinct prime factors ω(n) of an integer n. The theorem states that for large x, the distribution of (ω(n) - ln ln n)/√(ln ln n) approaches a standard normal distribution as n ranges up to x, with mean ln ln x + M (where M is the Meissel–Mertens constant M = γ + ∑_p [ln(1 - 1/p) + 1/p]) and variance ln ln x; this result highlights the central role of γ (via M) in quantifying fluctuations in the arithmetic structure of integers.

Generalizations

Stieltjes constants

The Stieltjes constants \gamma_n for n = 0, 1, 2, \dots form a sequence that generalizes the \gamma = \gamma_0, appearing as the coefficients in the Laurent series expansion of the \zeta(s) around its simple pole at s = 1: \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{\gamma_n}{n!} (s-1)^n. This expansion captures the regular part of \zeta(s) near s=1, with the constants defined by \gamma_n = \lim_{m \to \infty} \left( \sum_{k=1}^m \frac{(\ln k)^n}{k} - \frac{(\ln m)^{n+1}}{n+1} \right). Numerical computation of the Stieltjes constants relies on accelerated series representations or integral methods derived from the zeta function, enabling high-precision evaluations. For instance, \gamma_1 \approx -0.07281584548, \gamma_2 \approx -0.009690363192, and \gamma_3 \approx 0.002053834420; more broadly, values up to \gamma_{200} and beyond have been computed to thousands of decimal places by 2025 using efficient zeta function algorithms and asymptotic approximations for large n. The Stieltjes constants exhibit complex behavior, with their signs starting as \gamma_0 > 0, \gamma_1 < 0, \gamma_2 < 0, \gamma_3 > 0 before showing more irregular patterns for higher indices, and their magnitudes grow super-factorially with n. These constants connect to the distribution of prime numbers through explicit formulas, such as those expressing the Chebyshev function \psi(x) in terms of sums over zeta zeros, where higher-order terms involve the \gamma_n. In applications, the Stieltjes constants refine the prime number theorem by contributing to higher-order asymptotic expansions of \psi(x) - x, providing corrections beyond the leading term and improving error estimates in prime counting functions. They also play a role in studies of the Riemann zeta function's zeros, appearing in relations between derivatives at s=1 and sums over non-trivial zeros, which aids in numerical verification of hypotheses like the Riemann hypothesis.

Euler–Lehmer constants

The Euler–Lehmer constants \gamma(a, q) are defined for a positive q and an a with $1 \le a \le q and \gcd(a, q) = 1 by the limit \gamma(a, q) = \lim_{N \to \infty} \left( \sum_{\substack{1 \le m \le N \\ m \equiv a \pmod{q}}} \frac{1}{m} - \frac{\ln N}{q} \right). This limit exists and is finite. These constants generalize the Euler–Mascheroni constant \gamma by restricting the harmonic sum to terms in the with common difference q and initial residue a coprime to q. The modulus-dependent Euler–Lehmer constant \gamma_q is the of the \gamma(a, q) over the \phi(q) values of a coprime to q: \gamma_q = \frac{1}{\phi(q)} \sum_{\substack{1 \le a \le q \\ \gcd(a, q) = 1}} \gamma(a, q). A for the individual constants is \gamma(a, q) = -\frac{1}{q} \left( \psi\left( \frac{a}{q} \right) + \ln q \right), where \psi denotes the . As q \to \infty, \gamma_q \to \gamma. Values of \gamma(a, q) and \gamma_q have been computed to high precision for small moduli; for example, Lehmer provided tables for q \le 30, and subsequent computations extend to larger q up to several thousand as of 2025. Representative values include \gamma_1 = \gamma \approx 0.577216, \gamma_2 \approx 0.635181, and \gamma_3 \approx 0.37551. For prime q = p, the constant \gamma_p appears in the expansion of the of the \mathbb{Q}(\zeta_p) around s = 1, relating it to the for that field via products of L(1, \chi) over characters modulo p. The Euler–Lehmer constants arise in the asymptotics of partial sums of Dirichlet L-functions at s = 1 and in methods, where sums over progressions provide key estimates for the of primes and other arithmetic functions. Their arithmetic nature, including questions of , remains an active area of research; for instance, for fixed q > 1, at most one \gamma(a, q) can be algebraic. Like the Stieltjes constants, they connect to expansions involving L-functions.

Masser–Gramain constant

The Masser–Gramain constant, denoted δ, is a two-dimensional analogue of Euler's constant γ, emerging from the study of entire functions that map the ring of Gaussian integers ℤ to itself. Introduced by David W. Masser in 1979, it quantifies the asymptotic distribution of Gaussian integers within disks centered at the origin in the . This constant plays a key role in determining the growth rates of such arithmetic entire functions and has implications for transcendence theory, particularly in establishing lower bounds for measures related to numbers like e^π. Formally, δ is defined as \delta = \lim_{n \to \infty} \left( \sum_{k=2}^{n} \frac{1}{\pi r_k^2} - \ln n \right), where r_k denotes the radius of the smallest disk centered at 0 that contains exactly k nonzero Gaussian integers. This limit captures the "defect" in the area growth of these disks compared to the expected π r_k^2 ≈ k from , mirroring how γ arises from the harmonic series as a logarithmic defect in one . The of the limit follows from the equidistribution properties of the Gaussian lattice, and δ reflects the arithmetic structure of ℤ. François Gramain extended this work in 1981–1982, proving that non-constant entire functions f: ℂ → ℂ with f(ℤ) ⊆ ℤ must satisfy a growth condition lim sup (ln M(r))/r ≥ π/(2e) ≈ 0.5778, where M(r) is the maximum modulus; this bound is optimal and ties directly to transcendence results via comparisons with . Numerical evaluations confirm δ ≈ 1.81978, with rigorous bounds established as 1.819776 < δ < 1.819833 using advanced lattice point enumeration techniques. These computations, which require identifying r_k for large k (up to n ≈ 10^6 in the original work, with potential extensions to higher values), disprove an earlier conjecture by Gramain positing δ > 1.82 and rely on efficient algorithms for counting Gaussian integers inside circles, often accelerated by and verified enclosures. No higher-precision records beyond these bounds appear in the literature as of 2025, though the methods suggest potential for further refinement via optimized or arithmetic geometry tools for lattice counting. The of δ remains open, akin to γ, but its value informs bounds in Baker's theory of linear forms in logarithms, aiding transcendence proofs for expressions like e^{π √d} by constraining auxiliary function growth in several variables.