ABSTRACT. Let A = {a 1 < a 2 < a 3 ..... < a n < ...} be an infinite sequence of integers and let R 2 (n) = |{(i, j) :is small then L 1 norm of S + k k is large.

Let E be an elliptic curve defined over rational field Q and N be a positive integer. Now M E (N ) denotes the number of primes p, such that the group E p (F p ) is of order N . We show that M E (N ) follows Poisson distribution when an average is taken over a large class of curves.

ABSTRACT. In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions F and G under consideration are close to two nicely behaved functions A and B, such that the average value of A(n − h)B(n) over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of K(N), where K(N)/ log N is the expected number of primes such that a random elliptic curve over rationals has N points when reduced over those primes.

Let E be an elliptic curve defined over Q and let N be a positive integer. Now, M E (N ) counts the number of primes p such that the group E p (F p ) is of order N . In an earlier joint work with Balasubramanian, we showed that M E (N ) follows Poisson distribution when an average is taken over a family of elliptic curve with parameters A and B where A, B N 2010 Mathematics Subject Classification 11G05 (primary), 11G20, 11N05 (secondary).

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