On the binary additive divisor problem. (English) Zbl 1404.11117
Proc. Steklov Inst. Math. 299, 44-49 (2017); translation in Tr. Mat. Inst. Steklova 299, 50-55 (2017).
The binary additive divisor problem is the study of the sum
\[
\sum_{n=1}^Md(n)d(n+f)=MT(M,f)+E(M,f),
\]
where \(d(n)\) stands for the number of positive divisors of \(n\), \(f\) is a positive integer, \(T(M,f)\) is an explicit main term, closely related to a certain quadratic polynomial in \(\ln M\) with coefficients depending on \(f\), and \(E(M,f)\) is a remainder term.
The authors first recall the historical development of this study, mentioning in particular the contributions [Y. Motohashi, Ann. Sci. Éc. Norm. Supér. (4), 529–572 (1994; Zbl 0819.11038)] and [T. Meurman, in Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland. Berlin: de Gruyter, 223–246 (2001; Zbl 0967.11039)], which give uniform estimates of \(E(M,f)\), the best ones to this day.
Next, having observed that Meurman’s statement is more precise than Motohashi’s, the authors revisit the work of the latter, and prove that it can be sharpened to yield precisely Meurman’s result. Their main contribution is a careful analysis of an integral used by Motohashi.
The authors first recall the historical development of this study, mentioning in particular the contributions [Y. Motohashi, Ann. Sci. Éc. Norm. Supér. (4), 529–572 (1994; Zbl 0819.11038)] and [T. Meurman, in Proceedings of the Turku symposium on number theory in memory of Kustaa Inkeri, Turku, Finland. Berlin: de Gruyter, 223–246 (2001; Zbl 0967.11039)], which give uniform estimates of \(E(M,f)\), the best ones to this day.
Next, having observed that Meurman’s statement is more precise than Motohashi’s, the authors revisit the work of the latter, and prove that it can be sharpened to yield precisely Meurman’s result. Their main contribution is a careful analysis of an integral used by Motohashi.
Reviewer: Michel Balazard (Marseille)
MSC:
| 11N37 | Asymptotic results on arithmetic functions |
| 11N75 | Applications of automorphic functions and forms to multiplicative problems |
Software:
DLMFReferences:
| [1] | Deshouillers, J.-M.; Iwaniec, H., An additive divisor problem, J. London Math. Soc., Ser. 2, 26, 1-14, (1982) · Zbl 0462.10030 |
| [2] | Estermann, T., Über die darstellungen einer zahl als differenz von zwei produkten, J. Reine Angew. Math., 164, 173-182, (1931) · Zbl 0001.20302 |
| [3] | Heath-Brown, D. R., The fourth power moment of the Riemann zeta function, J. London Math. Soc., Ser. 3, 38, 385-422, (1979) · Zbl 0403.10018 · doi:10.1112/plms/s3-38.3.385 |
| [4] | Ingham, A. E., Some asymptotic formulae in the theory of numbers, J. London Math. Soc., 2, 202-208, (1927) · JFM 53.0157.01 · doi:10.1112/jlms/s1-2.3.202 |
| [5] | Kim, H. H., Functoriality for the exterior square of GL_{4} and the symmetric fourth of GL_{2}, J. Am. Math. Soc., 16, 139-183, (2003) · Zbl 1018.11024 · doi:10.1090/S0894-0347-02-00410-1 |
| [6] | Kuznetsov, N. V., Petersson’s conjecture for cusp forms of weight zero and linnik’s conjecture. sums of Kloosterman sums, Mat. Sb., 111, 334-383, (1980) · Zbl 0427.10016 |
| [7] | Kuznetsov, N. V., Convolution of the Fourier coefficients of the Eisenstein-Maass series, Zap. Nauchn. Semin. LOMI, 129, 43-84, (1983) · Zbl 0496.10013 |
| [8] | Meurman, T., On the binary additive divisor problem, 223-246, (2001), Berlin · Zbl 0967.11039 |
| [9] | Motohashi, Y., Spectral mean values of Maass waveform \(L\)-functions, J. Number Theory, 42, 258-284, (1992) · Zbl 0759.11026 · doi:10.1016/0022-314X(92)90092-4 |
| [10] | Motohashi, Y., The binary additive divisor problem, Ann. Sci. Éc. Norm. Supér., Sér. 4, 27, 529-572, (1994) · Zbl 0819.11038 |
| [11] | NIST Handbook of Mathematical Functions, Ed. by F.W. J.Olver,D.W. Lozier, R.F. Boisvert, and C. W. Clarke (Cambridge Univ. Press, Cambridge, 2010). · Zbl 1198.00002 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
