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MRB constant

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The MRB constant, named after Marvin Ray Burns, is a mathematical constant for which no closed-form expression is known. It is not known whether the MRB constant is algebraic or transcendental, nor even whether it is rational or irrational. However, it has been computed to 3,000,000 digits of precision with no termination or obvious period of digits.

Marvin Ray Burns published his discovery of the constant in 1999. Before verifying with colleague Simon Plouffe that such a constant had not already been discovered or at least not widely published, Burns called the constant "rc" for root constant.[1] At Plouffe's suggestion, the constant was renamed Marvin Ray Burns' Constant, and then shortened to "MRB constant" in 1999.[2]

Definition

The MRB constant is related to the following oscillating divergent series

Its partial sums

are bounded within the closed interval

since , and where the MRB constant[3] is defined as

The MRB constant can be explicitly defined by the following infinite sums[4]

where , thus fulfilling the necessary and sufficient conditions for the convergence of an alternating series.


One should use acceleration methods when computing a numerical approximation of the MRB constant because it can be shown that one must sum a number in the order of iterations of to get accurate digits of the MRB Constant. However, using a convergence acceleration of alternating series algorithm of Cohen-Villegas-Zagier one can compute the first 60 digits in only 100 iterations.[5]

CMRB

The decimal expansion of the MRB constant is[6]

(A037077)

The simple continued fraction expansion of the MRB constant is

giving the sequence

{0, 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, 17, 1, 6, 4, 1, 3, 3, ...}

The first few convergents from the simple continued fraction expansion of the MRB constant are (see talk page for pattern investigation)

Limit (with an even number of terms) of the power tower of CMRB

where is the MRB constant and is the the power tower height.

Decimal expansion of limit (with an even number of terms) of the power tower of CMRB

A052110 Decimal expansion of limit (with an even number of terms) where is the MRB constant defined in A037077.

{4, 6, 1, 9, 2, 1, 4, 4, 0, 1, 6, 4, 4, 1, 1, 4, 4, 5, 4, 0, 8, 5, 8, 8, 6, 4, 2, 6, 1, 4, 1, 9, 4, 5, 7, 8, 6, 3, 5, 0, 2, 8, 2, 8, 0, 1, 3, 6, 4, 8, 8, 2, 2, 8, 4, 4, 3, 4, 1, 6, 2, 9, 2, 7, 3, 5, 8, 9, ...}

CMRB - 1

The decimal expansion of the MRB constant - 1 is

The simple continued fraction expansion of the MRB constant - 1 is

giving the sequence

{-1, 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, 17, 1, 6, 4, 1, 3, 3, ...}

The first few convergents from the simple continued fraction expansion of the MRB constant - 1 are

CMRB - 1/2

The Cesàro sum and the Levin u-transform sum give the midpoint of the two limit points

The decimal expansion of the MRB constant - is

The simple continued fraction expansion of the MRB constant - is

giving the sequence

{-1, 1, 2, 4, 1, 10, 20, 2, 2, 2, 7, 1, 2, 2, 1, 2, 4, 9, 1, 1, 2, 4, 4, 1, ...}

Integrated analog of the series

The integrated analog of the series is a complex-valued integral of oscillatory character[7]

Not convergent in the continuum limit at , the limit of the sequence of integrals with an integral difference in the upper limits exists.

Ultraviolet limit MI of the sequence of oscillatory integrals

The ultraviolet limit of the sequence of oscillatory integrals is defined as

and has been evaluated by Richard J. Mathar.[7]

Real and imaginary part

The decimal expansion of the real part of MI is

The simple continued fraction for real part of is

giving the sequence

{0, 14, 7, 1, 2, 1, 23, 2, 1, 8, ...}


The decimal expansion of the imaginary part of MI is

The simple continued fraction for imaginary part of is

giving the sequence

{-1, 3, 6, 13, 41, 112, 1, 25, 1, 1, ...}

Absolute value of MI

The decimal expansion of absolute value is

(A157852)

The simple continued fraction for absolute value of is

giving the sequence

{0, 1, 2, 4, 1, 24, 1, 4, 1, 2, 4, 4, 1, 8, ...}

Series minus integrated analog of the series

By analogy with the Euler–Mascheroni constant

we may define

CMRB - MI

Real part of CMRB - MI

The decimal expansion of the real part of is

The simple continued fraction for real part of is

giving the sequence

{0, 8, 1, 1, 5, 1, 1, 1, 1, 2, 1, 5, 5, 1, ...}

Imaginary part of CMRB - MI

The decimal expansion of the imaginary part of is

The simple continued fraction for imaginary part of is

giving the sequence

{0, 1, 2, 6, 13, 41, 112, 1, 25, ...}

Absolute value of CMRB - MI

The decimal expansion of is

The simple continued fraction for is

giving the sequence

{0, 1, 2, 3, 1, 2, 1, 5, 9, 1, 8, 29, ...}

Inquiry on Feb 28, 2013

It seems that using regularization the divergent series

.

Any help here would be great!

For all complex z, the upper limit point of s(n)= is the .
For all real a, the partial sums s(n)= are bounded so that their limit points form an interval [-1.+ +a, ] of length 1-a.

See also

Notes

  1. Burns, Marvin R. (1999-01-23). "RC". math2.org. Retrieved 2009-05-05. 
  2. Plouffe, Simon (1999-11-20). "Tables of Constants". Laboratoire de combinatoire et d'informatique mathématique. Retrieved 2009-05-05. 
  3. Burns, Marvin R. (1999-09-01). "MRB Constant". Plouffe's Inverter. Retrieved 2009-05-05. 
  4. Weisstein, Eric W., MRB Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/MRBConstant.html]
  5. Marvin Ray Burns, The Geometry of the MRB constant, 2010.
  6. Marvin Ray Burns, 299998 digits (MRB constant), computed Aug 13, 2010 to Sept 8, 2010.
  7. 7.0 7.1 Mathar, Richard J. (2009). "Numerical evaluation of the oscillatory integral over exp(i*pi*x) x^(1/x)...". arΧiv:0912.3844. 

External links