Exploring Euler’s Constant Julian Havil Julian Havil princeton university press princeton and oxford Copyright c by Princeton University Press Published. J. Havil, Gamma, Exploring Euler’s Constant, Princeton University Press, Princeton and Oxford, , page G. Boros and V. Moll, Irresistible Integrals: . It was first defined by Euler (), who used the letter C and stated that it was ” worthy of serious consideration” (Havil , pp. xx and 51). The symbol gamma .

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The Euler-Mascheroni constantsometimes also called ‘Euler’s constant’ or ‘the Euler constant’ but not to be confused with the constant is defined as the limit of the sequence.

It was first defined by Eulerwho used the letter and stated that it was “worthy of serious consideration” Havilpp. The symbol was first used by Mascheroni It is not known if this constant is irrationallet alone transcendental Wellsp. The famous English mathematician G. Hardy is alleged to have offered to give up his Savilian Chair at Oxford to anyone who proved to be irrational Havilp.

Hilbert mentioned the irrationality of as an unsolved problem that seems “unapproachable” and in front of which mathematicians stand helpless Havilp. Hail and Guy are “prepared to bet that it is transcendental,” although they do not expect a proof to be achieved within their lifetimes.

If is a simple fractionthen it is known that Brent ; Wellsp. Papanikolaou to Havilp.

## Chebyshev Functions

The Euler-Mascheroni constant continued fraction is haivl by [0, 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, The Engel expansion of is given by 2, 7, 13, 19, 85,, Whittaker and Watsonp.

Integrals that give in combination with other simple constants include. Sondow; Borwein et al.

An interesting analog of equation 10 is given by. By taking the logarithm of both sides, an explicit formula for havio obtained. Gourdon and Sebahp. VaccaGerstwhere lg is the logarithm to base 2 and is the floor function. Nielsen earlier gave a series equivalent to 24.

## Rosser’s Theorem

Gosperwith replacing the undefined ; Bailey and Crandall can be obtained from equation 24 by rewriting as a double seriesthen applying Euler’s series transformation to javil of these series and adding to get equation Here, is a binomial coefficientand rearranging the conditionally convergent series is permitted because the plus and minus terms can first be havi in pairs, the resulting series of positive numbers rearranged, and then the series ungrouped back to plus and minus terms.

To see the equivalence, expand in a geometric seriesmultiply byand integrate termwise Sondow and Zudilin Other series for include. A rapidly converging limit for is given by. Another connection with the primes was provided by Dirichlet’s proof that the average number of divisors of all numbers from 1 to is asymptotic to.

Conway and Guy An elegant identity for is given by.

This gives an efficient iterative algorithm for by computing. Infinite products involving also arise from the Barnes G-function with positive integer. The cases and give. The symbol is sometimes also used for. Another proof of product 55 as well as an explanation for the resemblance between this product and the Wallis formula -like “faster product for “. Guillera hvil SondowSondowis given in Sondow This resemblance which is made even clearer by changing in Both these formulas are also analogous to the product for given by.

The values obtained after inclusion of the first terms of the product for are plotted above. A curious sum limit converging to is given by. Eine historisch-analytisch zusammenfassende Studie. Plausible Reasoning in the 21st Century.

A K Peters, Computational Paths to Discovery.

### Riemann Zeta Function Zeros — from Wolfram MathWorld

Annales de la Soc. Bruxelles 22, Monthly, Higher Transcendental Functions, Javil. Cambridge University Press, pp.

Monthly 76, Item in Beeler, M. Tables of Integrals, Series, and Products, 6th ed. A Foundation for Computer Science, 2nd ed. Princeton University Press, Adnotationes ad calculum integralem Euleri, Vol.

Ticino, Italy, and Reprinted in Euler, L. Leonhardi Euleri Opera Omnia, Ser. Monthly, a. Monthly, b. The Mathematica GuideBook for Programming. A Course in Modern Analysis, 4th ed. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions.

Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: Mon Dec 31 Euler-Mascheroni Constant The Euler-Mascheroni constantsometimes also called ‘Euler’s constant’ or ‘the Euler constant’ but not to be confused with the constant is defined as the limit of the sequence.

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