A brief contribution to the question of the divisor problem. (English. Russian original) Zbl 0798.11036

Math. Notes 53, No. 4, 454-457 (1993); translation from Mat. Zametki 53, No. 4, 148-152 (1993).
Let \(\alpha_ k\), as usual, denote the optimal exponent in the generalized divisor problem, so that \[ \sum_{n \leq x} d_ k(n) = x p_ k (\log x) + O(x^{\alpha_ k + \varepsilon}). \] It was shown by A. A. Karatsuba [Izv. Akad. Nauk SSSR, Ser. Mat. 36, 475-483 (1972; Zbl 0239.10025)] that \(\alpha_ k \leq 1-ck^{-2/3}\) \((k \geq k_ 0)\) with \(c = {1 \over 2} (2D)^{-2/3}\), where \(D\) is any constant for which the estimate \[ \zeta (\sigma + it) \ll t^{D(1-\sigma)^{3/2}} \log t\quad(t \geq 2,\quad {\textstyle{1 \over 2}} \leq \sigma \leq 1) \] holds. The main result of the paper is that any \(c<{1 \over 2} D^{-2/3}\) is admissible.
Unfortunately this same observation has already been made by A. Ivić [The Riemann zeta-function, p. 356 (Wiley, New York) (1985; Zbl 0556.10026)]. Indeed it appears to the reviewer that any \(c<({3 \over 2} D)^{-2/3}\) must be allowable. To prove this, one uses Theorem 7.9(A) of E. C. Titchmarsh [The theory of the Riemann zeta-function (Oxford) (1986; Zbl 0601.10026)] to establish the bound \[ \sigma_ m \leq 1 - (3D + \varepsilon)^{-2/3} m^{-2/3} (m \geq m_ \varepsilon) \] (in Titchmarsh’s notation), using induction on \(m\). One then has only to observe that \(\alpha_{2m}\), \(\alpha_{2m-1} \leq \sigma_ m\).


11N37 Asymptotic results on arithmetic functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
Full Text: DOI


[1] P. Dirichlet, Abh. Akad. Wiss. Berlin, 49-66 (1849).
[2] E. Landau, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., 687-771 (1912).
[3] G. F. Voronoi, Collected Works in Three Volumes, Vol. 2 [in Russian], Izd. Akad. Nauk UkrSSR, Kiev (1952).
[4] G. H. Hardy and J. E. Littlewood, Proc. London Math. Soc., Ser. 2,21, 39-74 (1922). · JFM 48.1163.01
[5] D. R. Heath-Brown, Recent Progress in Analytic Number Theory, Symposium, Durham 1979, Vol. 1, Academic Press, London (1981).
[6] A. Ivic, Acta Arithm.,52, No. 3, 241-253 (1989).
[7] H. E. Richert, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., 17-75 (1960).
[8] A. A. Karatsuba, Izv. Akad. Nauk SSSR, Ser. Mat.,36, No. 3, 475-483 (1972).
[9] E. I. Panteleeva, Mat. Zametki,44, No. 4, 494-505 (1988).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.