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Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
Applications of the harmonic series and its partial sums include Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are needed to provide a complete range of responses, the connected components of random graphs, the block-stacking problem on how far over the edge ...
In the 1980s Guy Robin showed [13] that the Riemann hypothesis is equivalent to the assertion that the following inequality is true for all n > 5040: (where γ is the Euler–Mascheroni constant) σ ( n ) < e γ n log log n ≈ 1.781072418 n log log n {\displaystyle \sigma (n)<e^{\gamma }n\log \log n\approx 1.781072418n\log \log ...
Meet the Euler-Mascheroni constant 𝛾, which is a lowercase Greek gamma. It’s a real number, approximately 0.5772, with a closed form that’s not terribly ugly; it looks like the image above.
The Euler–Mascheroni constant γ: In 2010 it has been shown that an infinite list of Euler-Lehmer constants (which includes γ/4) contains at most one algebraic number. [51] [52] In 2012 it was shown that at least one of γ and the Gompertz constant δ is transcendental. [53]
The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant. In mathematics , the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function :
Schanuel's conjecture would solve the first of these problems somewhat as it deals with algebraic independence and would indeed confirm that e + π is transcendental. It still revolves around the exponential function, however, and so would not necessarily deal with numbers such as Apéry's constant or the Euler–Mascheroni constant.