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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, transcendental, or even irrational.
The numerical value of MRB constant, truncated to 6 decimal places, is
The MRB constant is related to the following divergent series:
Its partial sums
The MRB constant can be explicitly defined by the following infinite sums:
Marvin Ray Burns published his discovery of the constant in 1999. The discovery is a result of a "math binge" that started in the spring of 1994. 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. At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999.
- Weisstein, Eric W. "MRB Constant". MathWorld.
- Mathar, Richard J. "Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity". arXiv:0912.3844.
- Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). PSI Press. Archived from the original (PDF) on April 30, 2013. Retrieved 16 January 2015.
- (sequence A037077 in the OEIS)
- (sequence A160755 in the OEIS)
- (sequence A173273 in the OEIS)
- Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.
- Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
- Burns, Marvin. "mrburns". Simeon Plouffe. Retrieved 12 January 2015.
- Burns, Marvin R. (12 April 2002). "Captivity's Captor: Now is the Time for the Chorus of Conversion". Indiana University. Retrieved 5 May 2009.
- Burns, Marvin R. (23 January 1999). "RC". math2.org. Retrieved 5 May 2009.
- Plouffe, Simon (20 November 1999). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 5 May 2009.