On a conjecture of Erdős and certain Dirichlet series

Abstract

ABSTRACT. Let f : Z/qZ → Z be such that f (a) = ±1 for 1 ≤ a < q, and f (q) = 0. Then Erdös conjectured that n≥1 f (n) n = 0. For q even, it is easy to show that the conjecture is true. The case q ≡ 3 ( mod 4) was solved by Murty and Saradha. In this paper, we show that this conjecture is true for 82% of the remaining integers q ≡ 1 ( mod 4).

Non-vanishing of Dirichlet series with periodic coefficients

“…We also revisit the approach of Baker, Birch and Wirsing. We are hopeful that this approach will find new applications such as in the study of the folklore Erd ös conjecture discussed in [3].…”

Transcendental numbers and special values of Dirichlet series

“…Using some algebraic number theory, Murty and Saradha [38] settled Erdös's conjecture if q ≡ 3 (mod 4). If q ≡ 1 (mod 4), the conjecture is still open, though it was shown by Chatterjee and Murty [11] that the conjecture is true for at least 82 percent of q with q ≡ 1 (mod 4). Most likely, one needs to understand the arithmetic significance of the non-vanishing to settle the conjecture completely.…”

On Primitivity and Vanishing of Dirichlet Series

“…Baker's theorem is required in their proof as we can write L(1, f ) as a linear form in logarithm of cyclotomic numbers by appealing to Plancherel's theorem. Using this idea along with Theorem 1, Okada [15] gave a necessary and sufficient criterion for the vanishing of L(1, f ) and most of the attempts in understanding Erdös Conjecture are based on applying this criterion (see [6], [24], [25],…”

“…Conjecture involved solving only one equation : either it is finding the algebraic component α of α log u where u is a unit in Z[ζ N ] and prove α is non-zero by arithmetic considerations (like [14]) or use one of the linear equations given in Okada's condition to obtain a bound on the coefficients by taking absolute values or minor considerations (like [24], [25], [6]). To obtain better results, we use our criteria to solve a system of equations.…”