In this short note, we give a proof of the Riemann hypothesis for Goss v-adic zeta function ζv(s), when v is a prime of Fq[t] of degree one. The Riemann Hypothesis says that the non-trivial complex zeros of the Riemann zeta function all lie on a line Re(s) = 1/2 in the complex plane. An analog [7, 2, 5] of this statement was proved for the Goss zeta function for Fq[t], for q a prime by Wan [7] (see also [1]) and for general q by Sheats [4]. The proofs use the calculation of the Newton polygons associated to the power series these zeta functions represent, and their slopes are calculated or estimated by the degrees of power sums which make up the terms. In [1, 4], the exact degrees derivable (as noticed by Thakur) from incomplete work of Carlitz were justified and the Riemann hypothesis was derived. In this paper, we look at the v-adic Goss zeta function and prove analog of the Riemann hypothesis, in case where v is a prime of degree one in Fq[t], using the valuation formulas for the corresponding power sums given essentially (see below) in [6] and following the method of [1], [5, Sec. 5.8]. We note that for q = 2, this was already shown by Wan [7] using the earlier calculation at the infinite place. See below for the details. We also note that the situation for higher degree v is different [6, Sec. 10] and is not fully understood. We now give the relevant definitions and describe the results precisely. In the function field number field analogy, we have A = Fq[t], K = Fq(t), K∞ = Fq((1/t)) and C∞, the completion of algebraic closure of K∞ as analogs of Z, Q, R and C respectively. We consider v-adic situation where v is an irreducible polynomial of A, and Kv the completion of K at v, as analog of p-adic situation, where p is a prime in Z and Qp the field of p-adic numbers. For v a prime of Fq[t], the v-adic zeta function of Goss is defined on the space Sv := Cv × lim ←− j Z/(q v − 1)pZ, where Cv is the completion of an algebraic closure of Kv and the lim ←− j Z/(q v − 1)pZ is isomorphic to the product of Zp with the cyclic group Z/(q v − 1)Z. See [2, §8.3], [5, §5.5(b)] for motivation and details. 1 ar X iv :1 40 8. 11 11 v1 [ m at h. N T ] 5 A ug 2 01 4 For s = (x, y) ∈ Sv, the Goss v-adic zeta function is then defined as