Functiones et Approximatio Commentarii Mathematici

Formal proofs of degree 5 binary BBP-type formulas

Kundle Adegoke

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We study the analytic behavior of a power series with coefficients containing the von Mangoldt function. In particular, we extend an explicit formula of Hardy and Littlewood for related functions and derive further representation formulas in the unit disk that reveal logarithmic singularities on a dense subset of the unit circle. As an essential tool for proving the square integrability of occurring limit functions together with respective error estimates we contribute a new proof of a Ramanujan-like expansion of an arithmetic function consisting of the von Mangoldt function and the Euler function.

Article information

Funct. Approx. Comment. Math., Volume 48, Number 1 (2013), 19-27.

First available in Project Euclid: 25 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11Y60: Evaluation of constants
Secondary: 30B99: None of the above, but in this section

BBP type formulas digit extraction formulas polylogarithm constants


Adegoke, Kundle. Formal proofs of degree 5 binary BBP-type formulas. Funct. Approx. Comment. Math. 48 (2013), no. 1, 19--27. doi:10.7169/facm/2013.48.1.2.

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  • D.H. Bailey, P.B. Borwein, and S. Plouffe, On the rapid computation of various polylogarithmic constants, Mathematics of Computation 66(218) (1997), 903–913.
  • D.H. Bailey, A compendium of bbp-type formulas for mathematical constants, October 2010, http:/\!/ dhbailey/dhbpapers/bbp-formulas.pdf.
  • D.H. Bailey and R.E. Crandall, On the random character of fundamental constant expansions, Experimental Mathematics 10 (2001), 175.
  • J.M. Borwein, D. Borwein, and W.F. Galway, Finding and excluding $b$-ary machin-type BBP formulae, 2002, available online at http:/\!/ dhbailey/dhbpapers/machin.pdf.
  • M. Chamberland, Binary bbp-formulae for logarithms and generalized gaussian-mersenne primes, Journal of Integer Sequences 6, 2003.
  • H.R.P. Ferguson, D.H. Bailey, and S. Arno, Analysis of pslq, an integer relation finding algorithm, Math. Comput. 68 (1999), 351–369.
  • D.H Bailey, Algorithms for Experimental Mathematics I, 2006, available online at http:/\!/ dhbailey/dhbpapers.
  • D.J. Broadhurst, Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $\zeta(3)$ and $\zeta(5)$, arXiv:math/9803067v1 [math.CA], 1998, available online at http:/\!/
  • K. Adegoke, A non-pslq route to bbp-type formulas, Journ. Maths. Res. 2 (2010), 56–64.
  • K. Adegoke, Symbolic routes to BBP-type formulas of any degree in arbitrary bases, App. Math. Inf. Sci. 5(2) (2011), 264–275.
  • K. Adegoke, A novel approach to the discovery of ternary BBP-type formulas for polylogarithm constants, Notes Num. Theory and Disc. Math. 17(1) (2011), 4–20.
  • K. Adegoke, New Binary and Ternary Digit Extraction (BBP-type) Formulas for Trilogarithm Constants, New York J. Math. 16 (2010), 361–367.
  • K. Adegoke, A novel approach to the discovery of binary BBP-type formulas for polylogarithm constants, Integers 12 (2012), 345–371.