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Calculating a determinant associated with multiplicative functions

P. CodecáM. Nair — 2002

Bollettino dell'Unione Matematica Italiana

Let h be a complex valued multiplicative function. For any N N , we compute the value of the determinant D N := det i | N , j | N h i , j i j where i , j denotes the greatest common divisor of i and j , which appear in increasing order in rows and columns. Precisely we prove that D N = p l N 1 p l l + 1 i = 1 l h p i - h p i - 1 τ N / p l . This means that D N 1 / τ N is a multiplicative function of N . The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions f n , with 0 f p < 1 , as minimal values of certain...

Links between Δ x , N = n x N , n , N = 1 1 - x ϕ N and character sums

P. CodecáM. Nair — 2003

Bollettino dell'Unione Matematica Italiana

We express Δ x , N , as defined in the title, for x = a q and q prime in terms of values of characters modulo q . Using this, we show that the universal lower bound for Δ N = sup x R Δ x , N can, in general, be substantially improved when N is composed of primes lying in a fixed residue class modulo q . We also prove a corresponding improvement when N is the product of the first s primes for infinitely many natural numbers s .

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