Exponential Sums Involving Arithmetic Functions And Prime Numbers In Arithmetic Progressions

The study of exponential sums is a central topic in number theory and it has important theoretical significance and practical value.Let M denote theclass of complex valued multiplicative functions,and let M1(?)M be the set of those multiplicative functions f for which |f(n)| ?1 holds for every natural number n.Given any irrational number ?,let e(?)= e2?i? Let M' denote the set of arithmetic functions which satisfy special conditions,for example M'? M1.For f ? M1 or f being some special arithmetic function,this article mainly studies the asymptotic behavior of averages of the following form:where E is an integral set andIn 1954,Vinogradov[25]proved a well-known result that if Q(n)= ?knk+?k-1nk-1+…+?1n is a polynomial with real coefficients and such that at least one among ?k,,…,?1 is an irrational number,thenIn 1974,Daboussi and Delange[6]proved that given any irrational number? and f ? M1 the following is rightIn 1986,I.Katai[9]proved thatAfter that,Delange extended the result for f ? L2,that is for those f ? M which satisfy Indlekofer also extended the result for f ?L*,that is for any f ? M which satisfiesIn 2012,J.M.De Koninck and I.Katai[17]defined a new arithmetic function l(n):g1[F1(n))…gs(Fs(n)).And F1(x),…,Fs(x)? Z[x]take only positive values at positive arguments.And gi {i=1,2,…,s)are all complex-valued multiplicative functions and satisfy special conditions.Let they proved the following results:In this paper we will consider such types of sums but on shifted primes in arithmetic progressions.Given integers k,l which satisfy(k,l)= 1,we will prove the following result:where l(n):g1(F1(n))…g(Fs(n))and F1(x),…,Fs(x)? Z[x]take only positive values at positive arguments.For all i = 1,2,…,s,g1 are all complex-valued multiplicative functions and satisfy special conditions(there is intro-duction in the third section).We use the classical method of Turan—Kubilius inequality[24]to prove our result.We also use the third chapter in G.Tenenbaum[24],the related lemmas in J.M.De Koninck and I.Katai[17]and the basic knowledge in number theory.There are three sections in this paper.In the first section,we mainly introduce the background and results of the former researchers.In the second section,we will do some preparatory work.Finally,we will introduce some lemmas and prove the main theorem.