Graphical Representations of Depleted Zeta Subseries

base

10

length of string L

1

digit 1

δ

1

0

digit 2

δ

2

0

digit 3

δ

3

0

digit 4

δ

4

0

digit 5

δ

5

0

partial sums

plot comparison

partial sums of Ψ(1,0,10) up to n

It is well-known that the harmonic series diverges; equivalently,

ζ(1)

equals infinity. Remarkably, modified forms of

ζ

, denoted here by

Ψ

, can yield various converging values. For example, deleting all terms in the harmonic series whose denominator contains a 9 converges to approximately 23; this is known as the Kempner sum. More generally,

Ψ(s,n,b)

represents a subseries of

ζ(s)

with terms deleted whose denominator's base

b

representation contains the string of digits

n

given by

δ

1

…

δ

L

.