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The divisor problem for arithmetic progressions with small modulus. (English) Zbl 0726.11056

Let \(d_{\ell}(n)\) denote the number of ordered \(\ell\)-tuples of positive integers whose product is equal to n. The divisor problem is that of writing \[ D_{\ell}(x;a,q):=\sum_{n\leq x;\quad n\equiv a mod q}d_{\ell}(n) \] in “main term plus remainder” form; the following results are obtained: Theorem 1 gives an expression for the main term in terms of Laurent coefficients of Hurwitz zeta functions and functions which are antiderivatives of powers of log x. The expression obtained is similar to one given in a paper of A. F. Lavrik [Proc. Steklov Inst. Math. 142, 175-183 (1979); transl. from Tr. Mat. Inst. Steklova 142, 165-173 (1976; Zbl 0413.10035)] when no arithmetic progression is involved. In Theorem 2 a bound for the remainder term is obtained when \(\ell \geq 4\) and \(k:=(a,q)=1\), and Proposition 3 extends the result to the case \(k>1\), showing that the remainder term is (roughly) on the order of \[ (\frac{x}{k})^{(\ell -1)/(\ell +2)}+(\frac{x}{k})^{1/2}(\frac{q}{k})^{(\ell -4)/4}. \] The proof of Theorem 1 is essentially a residue computation, and that of Theorem 2 follows the argument of §12.3 of Titchmarsh’s book on the zeta function; both involve the generating function \(\sum_{n\equiv a mod q}d_{\ell}(n)/n^ s\), which can be written in terms of Hurwitz zeta functions (for computing the main term) or L-functions (for estimating the error term). Proposition 3 follows from a similar technique given by D. R. Heath-Brown in the case \(\ell =3\) [Acta Arith. 47, 29-56 (1987; Zbl 0549.10034)].

MSC:

11N37 Asymptotic results on arithmetic functions
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