The Erdős-Borwein constant 
, sometimes also denoted 
, is the sum of the reciprocals
of the Mersenne numbers,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
(OEIS A065442), where 
is the divisor
function and 
is a q-polygamma
function. The transformation from equation (1) to (2) follows from the series
transformation
 |
(7)
|
due to Clausen (Knuth 1998, pp. 155 and 157), valid for 
, with 
here.
Erdős (1948) showed that the constant 
is irrational. Borwein
(1992) subsequently showed that
 |
(8)
|
is irrational for an integer 
with 
and a nonzero rational number 
satisfying 
for every positive integer 
.
Lambrinoudis (2026) reported that 
is not a quadratic irrational
number. More generally, for every integer 
, the numbers 1, 
, and 
are linearly independent
over 
,
where
 |
(9)
|
This does not establish transcendence. The author credits generative AI with developing the argument and Lean formalization under his supervision. As of Oct. 15, 2026, the project reported a complete formalization with no additional mathematical assumptions, but independent specialist review had not been reported.
Bailey and Crandall (2002, §5, p. 540) asked whether every finite binary word occurs in the binary
expansion of 
, a property they called 2-density. Campbell (2026) proved
that the word 11 occurs infinitely often. The authorless
"Disjunctivity of the Erdős-Borwein Constant" (2026) claims that,
for every fixed binary word 
, there is a constant 
such that
 |
(10)
|
for all sufficiently large 
, where 
counts occurrences beginning among the first 
binary digits. Thus the claimed
lower bound implies that every finite binary word
occurs infinitely often in the binary expansion
of 
.
The accompanying Lean development checks the qualitative conclusion conditional on two explicit hypotheses on the distribution of prime numbers, but does not formalize the displayed quantitative lower bound. As of Sep. 8, 2026, no specialist review had been reported. VibeMathed (2026) classifies the work as AI-discovered and attributes it to GPT-6 Astra, but the released source does not identify an author or state an AI role.
