Distribution and non-vanishing of special values of L-series attached to Erdős functions

Abstract

In a written correspondence with A. Livingston, Erdős conjectured that for any arithmetical function f , periodic with period q, taking values in {−1, 1} when q ∤ n and f (n) = 0 when q n, the series ∑ ∞ n=1 f (n) n does not vanish. This conjecture is still open in the case q ≡ 1 mod 4 or when 2φ(q) + 1 ≤ q. In this paper, we obtain the characteristic function of the limiting distribution of L(k, f ) for any positive integer k and Erdős function f with the same parity as k. Moreover, we show that the Erdős conjecture is true with "probability" one.2010 Mathematics Subject Classification. 11M99.

On Primitivity and Vanishing of Dirichlet Series

“…In this proof, we assume that L(1, g) = 0 and therefore the components of g satisfy (V D ) for all divisors D of pN. Splitting g d accordingly and choosing D co-prime to p, we get functions F D and G D such that they satisfy (18). P(F D ) and P(G D ) will be certain linear combinations of elements in…”

“…where n is a representative of a residue mod pN/D satisfying n ≡ 1 mod p and n ≡ n mod N/D. Arriving at (18) for divisors D of N: Now, we consider (V pD ) for the same divisor…”

“…Following [18], we describe the functions f satisfying the condition of Conjecture 1 as Erdös functions. We abbreviate the infinite sum as L(1, f ) where L(s, f ) denotes the Dirichlet series ∞ n=1 f (n)n −s for ℜs > 1.…”