We express Delta(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 Delta(N)= sup|Delta(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.
Links between $\Delta(x,N)=\sum\Sb n\leq xN\\(n,N)=1\endSb 1-x\phi(N)$ and character sums.
CODECA', Paolo;NAIR, Mohan K.
2003
Abstract
We express Delta(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 Delta(N)= sup|Delta(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.File in questo prodotto:
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