Emre Alkan

Lehmer k-tuples

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

ARITHMETICAL FUNCTIONS IN SEVERAL VARIABLES

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).

Averages of values of L-series

Abstract. We obtain an exact formula for the average of values of L-series over two independent odd characters. The average of any positive moment of values at s = 1 is then expressed in terms of finite cotangent sums subject to congruence conditions. As consequences, bounds on such cotangent sums, limit points for the average of first moment of L-series at s = 1 and the average size of positive moments of character sums related to the class number are deduced.

RAMANUJAN SUMS AND THE BURGESS ZETA FUNCTION

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis.

NONVANISHING OF THE RAMANUJAN TAU FUNCTION IN SHORT INTERVALS

We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals.

Diophantine approximation with arithmetic functions, I

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

Pair correlation of sums of rationals with bounded height

Abstract For each positive integer Q, let ℱ Q denote the Farey sequence of order Q. We prove the existence of the pair correlation measure associated to the sum ℱ Q + ℱ Q modulo 1, as Q tends to infinity, and compute the corresponding limiting pair correlation function.

Series representing transcendental numbers that are not U-numbers

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahlers classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apery type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

Davenport constant for finite abelian groups

Abstract For a finite abelian group G, we investigate the length of a sequence of elements of G that is guaranteed to have a subsequence with product identity of G. In particular, we obtain a bound on the length which takes into account the repetitions of elements of the sequence, the rank and the invariant factors of G. Consequently, we see that there are plenty of such sequences whose length could be much shorter than the best known upper bound for the Davenport constant of G, which is the least integer s such that any sequence of length s in G necessarily contains a subsequence with product identity. We also show that the Davenport constant for the multiplicative group of reduced residue classes modulo n is comparatively large with respect to the order of the group, which is φ(n),when n is in certain thin subsets of positive integers. This is done by studying the Carmichael’s lambda function, defined as the maximal multiplicative order of any reduced residue modulo n, along these subsets.