The elementary method of Balog and Ruzsa and the large sieve of Linnik are utilized to investigate the behaviour of the 𝐿 1 norm of an exponential sum over the primes. A new proof of a lower bound due to Vaughan for the 𝐿 1 norm of an exponential sum formed with the von Mangoldt function is furnished.

Let E/Q be an elliptic curve. For a prime p of good reduction, let r(E, p) be the smallest non-negative integer that gives the x-coordinate of a point of maximal order in the group E(F p ). We prove unconditionally that r(E, p) > 0.72 log log p for infinitely many p, and r(E, p) > 0.36 log p under the assumption of the Generalized Riemann Hypothesis. This can be viewed as an elliptic curve analogue of classical lower bounds on the least primitive root of a prime.

Let /Q be an elliptic curve. For a prime of good reduction, let ( , ) be the smallest non-negative integer that gives the -coordinate of a point of maximal order in the group (F ). We prove unconditionally that ( , ) > 0.72 log log for infinitely many , and ( , ) > 0.36 log under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime. IntroductionLet /F be an elliptic curve. Recall that there exist unique positive integers , such that (F ) Z/ Z × Z/ Z and | . Here is the maximal order of a point of (F ). In order to find a point on /F of maximal order, a natural strategy is to compute the orders of points with -coordinates 0, 1, 2, . . . and continue until the desired point is found. In practice, this works fairly well. A natural question is how long this process takes in the worst case.Along these lines, fix an elliptic curve /Q and let be a prime of good reduction. Let ( , ) denote the minimal -coordinate of a point of maximal order in the reduction /F . The goal of this note is to prove the following two lower bounds on ( , ). Theorem 1.1 Let /Q be an elliptic curve. There are infinitely many primes such that ( , ) > 0.72 log log . Theorem 1.2 Let /Q be an elliptic curve. Under GRH, there are infinitely many primes such that ( , ) > 0.36 log .These results can be viewed as elliptic curve analogues of lower bounds on the least primitive root ( ) of a prime . Pillai [14] proved that there is a positive constant such that ( ) > log log for infinitely many . Using Linnik's theorem in Pillai's proof, Fridlender [6] and Salié [15] improved the result to the following. We include a proof since it inspired the result of the present paper.

We study iterated differential polynomial rings over a locally nilpotent ring and show that a large class of such rings are Behrens radical. This extends results of Chebotar and Chen et al.2010 Mathematics Subject Classification. 16N40.

Let R be a Q-algebra and d be a locally nilpotent derivation on R. We will show that the Jacobson radical of a differential polynomial ring R[x; d] equals I[x; d] where I is a nil ideal of R. This answers a question posed by Agata Smoktunowicz.