## Functiones et Approximatio Commentarii Mathematici

### On certain integrals involving the Dirichlet divisor problem

#### Abstract

We prove that $$\int_1^X\Delta(x)\Delta_3(x)dx \ll X^{13/9}\log^{10/3}X, \qquad \int_1^X\Delta(x)\Delta_4(x)\d x \ll_\varepsilon X^{25/16+\varepsilon},$$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\zeta^k(s)$ ($\Delta_2(x) \equiv \Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\Delta_k(x)$. We also obtain the analogues of the above bounds when $\Delta(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\zeta(1/2+it)|$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 62, Number 2 (2020), 247-267.

Dates
First available in Project Euclid: 26 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.facm/1572055509

Digital Object Identifier
doi:10.7169/facm/1819

Mathematical Reviews number (MathSciNet)
MR4113988

Zentralblatt MATH identifier
07225512

#### Citation

Ivić, Aleksandar; Zhai, Wenguang. On certain integrals involving the Dirichlet divisor problem. Funct. Approx. Comment. Math. 62 (2020), no. 2, 247--267. doi:10.7169/facm/1819. https://projecteuclid.org/euclid.facm/1572055509

#### References

• [BW] J. Bourgain and N. Watt, Mean square of zeta function, circle problem and divisor problem revisited, preprint available at arXiv:1709.04340.
• [Iv1] A. Ivić, The Riemann zeta-function, John Wiley & Sons, New York 1985 (reissue, Dover, Mineola, New York, 2003).
• [Iv2] A. Ivić, Large values of certain number-theoretic error terms, Acta Arith. 56 (1990), 135–159.
• [Iv3] A. Ivić, Mean values of the Riemann zeta function, Tata Institute of Fund. Research, LN's 82, (distributed by Springer Verlag, Berlin etc.), Bombay, 1991, to be found online at www.math.tifr.res.in/~publ/ln/tifr82.pdf.
• [Iv4] A. Ivić, Some problems on mean values of the Riemann zeta-function, Journal de Théorie des Nombres Bordeaux 8 (1996), 101–122.
• [Iv5] A. Ivić, On the mean square of the zeta-function and the divisor problem, Annales Acad. Scien. Fennicae Math. 32 (2007), 1–9.
• [Iv6] A. Ivić, On the divisor function and the Riemann zeta-function in short intervals, The Ramanujan Journal 19 (2009), 207–224.
• [IvZh] A. Ivić and W. Zhai, On some mean value results for the zeta-function and a divisor problem II, Indagationes Mathematicae 26 (2015), no. 5, 842–866.
• [Jut] M. Jutila, Riemann's zeta-function and the divisor problem, Arkiv Mat. 21 (1983), 75–96 and II, ibid. 31 (1993), 61–70.
• [KV] A.A. Karatsuba and S.M. Voronin, The Riemann zeta-function, Walter de Gruyter, Berlin–New York, 1992.
• [Kol] G. Kolesnik, On the estimation of multiple exponential sums. Recent progress in analytic number theory, Vol. 1 (Durham, 1979), pp. 231–246, Academic Press, London-New York, 1981.
• [LT] Y.-K. Lau and K.-M. Tsang, Mean square of the remainder term in the Dirichlet divisor problem, Journal de Théorie des Nombres de Bordeaux 7 (1995), 75–92.
• [Meu] T. Meurman, On the mean square of the Riemann zeta-function, Quarterly J. Math., Oxford II. Ser. 38 (1987), 337–343.
• [Tit] E.C. Titchmarsh, The theory of the Riemann zeta-function (2nd edition), Oxford University Press, Oxford, 1986.
• [Ton] K.-C. Tong, On divisor problems III, Acta Math. Sinica 6 (1956), 515–541.
• [Zh] W.-P. Zhang, On the divisor problem, Kexue Tongbao (English Ed.) 33 (1988), no. 17, 1484–1485.