Abstract.We consider the problem of the computation of inf p θp over the set of exponent pairs P p under linear constrains for a certain class of objective functions θ. An effective algorithm is presented. The output of the algorithm leads to the improvement and establishing new estimates in the various divisor problems in the analytical number theory.

Consider the operator E on arithmetic functions such that Ef is the multiplicative arithmetic function defined by (Ef )(p a ) = f (a) for every prime power p a . We investigate the behaviour of E m τ k , where τ k is a kdimensional divisor function and E m stands for the m-fold iterate of E. We estimate the error terms of n x E m τ k (n) for various combinations of m and k. We also study properties of E m f for arbitrary f and sufficiently large m.Our study provides a unified approach to functions with exponential divisors. We improve special cases of the Dirichlet asymmetric divisor problem and several results on the exponential divisor and totient functions.

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Abstract. Our paper is devoted to several problems from the field of modified divisors: namely exponential and infinitary divisors. We study the behaviour of modified divisors, sum-of-divisors and totient functions. Main results concern with the asymptotic behaviour of mean values and explicit estimates of extremal orders.

N. Minculete has introduced the divisor-of-order-r function τ (r) and the sum-of-divisors-of-order-r function σ (r) . We investigate the asymptotic behaviour of n≤x τ (r) (n) and n≤x σ (r) (n) and improve known estimates.