Lambert Series

A Lambert series is a series of the form

(1)

for whenever the series converges absolutely. Its power series expansion is

			(2)
			(3)

where

(4)

The particular case is sometimes denoted

			(5)
			(6)
			(7)

for (Borwein and Borwein 1987, pp. 91 and 95), where is a q-polygamma function. Special cases and related sums for include

			(8)
			(9)
			(10)
			(11)
			(12)

(Borwein and Borwein 1987, pp. 91-92), which arise in the reciprocal Fibonacci and reciprocal Lucas constants.

Some series of this type include

			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)

for , with for the expressions involving . Here is the Möbius function, is the totient function, is the number of divisors of , is the divisor function, is the sum of squares function counting representations , distinguishing signs and order, where and are integers (Hardy and Wright 1979). Also, is a Jacobi theta function (Bailey et al. 2006), is the Liouville function, and is the least significant bit of .

For the particular Lambert value

(23)

Lambrinoudis (2026) reported irrationality for every rational number with integers satisfying

(24)

For integer , the same work claims that 1, , and are linearly independent over , so these values are neither rational nor quadratic irrational numbers. The case is the Erdős-Borwein constant. The rational-base criterion does not include , for which irrationality remains open. The author credits generative AI with developing the mathematics and Lean formalization. As of Oct. 15, 2026, independent specialist review had not been reported.