Abstract. Generalizing a classical problem of Lehmer, in this paper we provide an asymptotic result for the number of Lehmer k-tuples.

Abstract. Let f = ∞ n=1 a f (n)q n be a cusp form with integer weight k ≥ 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1, letImproving on work of Balog, Ono, and Serre we show that i f (n) f,φ φ(n) for almost all n, where φ(x) is any good function (e.g. such as log log(x)) monotonically tending to infinity with x. Using a result of Fouvry and Iwaniec, if f is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all n that i f (n) f,ε n 69 169 +ε . We also obtain conditional results depending on the Generalized Riemann Hypothesis and the Lang-Trotter Conjecture.

Motivated by the classical work of Ramanujan and recent work of Berndt and Zaharescu, we establish certain infinite families of identities relating values of Dirichlet L-functions at positive integers to Gauss sums and trigonometric sums twisted with characters.

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In this paper we investigate the ring Ar(R) of arithmetical functions in r variables over an integral domain R. We study a class of absolute values, and a class of derivations on Ar(R). We show that a certain extension of Ar(R) is a discrete valuation ring. We also investigate the metric structure of the ring Ar(R).