On a Conjecture of Livingston

Abstract

Abstract. In an attempt to resolve a folklore conjecture of Erdös regarding the nonvanishing at s = of the L-series attached to a periodic arithmetical function with period q and values in {− , }, Livingston conjectured theQ -linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite q ≥ , highlighting that a new approach is required to settle ErdösâĂŹs conjecture. We also prove that the conjecture is true for prime q ≥ , and indicate that more ingredients will be needed to settle ErdösâĂŹs conjecture for prime q.

A note on a variant of a conjecture of Rohrlich

“…In fact for the case when we do not have the co‐prime condition in the above Proposition 2.3, Siddhi Pathak [17] gave the necessary and sufficient conditions for the set to be linearly independent over the field of algebraic numbers (for a proof see Theorems 1.1 and 1.2). Proposition Let $$q \geqslant 2$$ be an integer.…”

“…and hence by using equation ( 3) and (17), equation ( 16) can be rewritten as Observe that when 𝑞 does not satisfy any of the conditions given in Proposition 2.3, then the set in equation (19) itself is not linearly independent by using Proposition 2.…”

“…Since 𝑀 is linearly independent over ℚ, therefore we have 𝑑 𝑡 = 0 for all 𝑡 ∈ 𝐼. Thus by using equations (17) and (20), altogether we get a linear homogeneous system of at most 𝜙(𝑞)∕2 − 1 many equations in the variables 𝑐 , 𝑎 𝑠 with rational coefficients. Hence, we will get a non-trivial rational solution for the variables 𝑐 , 𝑎 𝑠 in equation (15).…”