Moments in the exact summation of the curious series of Kempner type

Abstract

We obtain for the Kempner series (i.e. harmonic series where certain digits are excluded from all denominators, for example the digit 9 in base 10) new representations as geometrically convergent series. The coefficients for these representations involve the power sums on the allowed digits and the moments of an explicitly described measure on the unit interval. These moments can be computed numerically by recurrence. We establish a priori lower and upper bounds for them. This allows converting the theoretical formulas into an efficient numerical algorithm.

15 pages. To appear (with better introduction, conclusion and style) in American Mathematical Monthly in 2025. Compared to v1, the definition of set curly A was slightly changed to handle digit zero allowed or not in a more unified way (the measure is exactly same as in v1). References to further works by the author and others got added

References in corpus (3)

• Dirichlet series of integers with missing digits
• Digamma function and general Fischer series in the theory of Kempner sums
• Summing the "exactly one 42" and similar subsums of the harmonic series

Cited by in corpus (1)

• Some series representing the zeta function for $\Re (s)>1$