Erdős Problem #1049
Let t>1 be a rational number. Is Σ_n=1^∞1/t^n-1=Σ_n=1^∞ τ(n)/t^n irrational, where τ(n) counts the divisors of n? A conjecture of Chowla.
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Open irrationality problems
Is a given series or constant irrational, or transcendental? Most problems here are Erdős's questions about series built from integer sequences; partial results and numerical evidence are both useful.
Erdős Problem #243
Let a_1 < a_2 < … be a sequence of integers such that lim_n→∞ a_n/a_n-1^2 = 1 and Σ 1/a_n ∈ ℚ. Then, for all sufficiently large n ≥ 1, a_n = a_n-1^2 - a_n-1 + 1.
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Erdős Problem #247
Let n_1 < n_2 < ⋯ be a sequence of integers such that limsup n_k/k = ∞. Is Σ_k=1^∞ 1/2^n_k transcendental?
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Erdős Problem #249
Is Σ_n φ(n)/2^n irrational? Here φ is the Euler totient function.
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Erdős Problem #251
Is Σ_n=1^∞ p_n/2^n irrational? Here p_n is the n-th prime (p_1=2, p_2=3, …).
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Erdős Problem #252
Erdős Problem 252: irrationality of the sum for a given k.
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Erdős Problem #257
Let A⊆ℕ be an infinite set. Is Σ_n∈ A 1/2^n - 1 irrational?
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Erdős Problem #263
Is a_n = 2^2^n an irrationality sequence in the above sense?
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Erdős Problem #264
Is n! an example of an irrationality sequence?
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Erdős Problem #68
Is Σ_n=2^∞ 1/n!-1 irrational?
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