## Tokyo Journal of Mathematics

### On the Fourth Moment of the Epstein Zeta Functions and the Related Divisor Problem

Keiju SONO

#### Abstract

In this paper, we study the fourth moment of the Epstein zeta function $\zeta (s;Q)$ associated to a $n\times n$ positive definite symmetric matrix $Q$ ($n\geq 4$) on the line $\mathrm{Re}(s)=\frac{n-1}{2}$. We prove that the integral $\int _{0}^{T}|\zeta (\frac{n-1}{2}+it;Q)|^{4}dt$ is evaluated by $O(T(\mathrm{log}\,T)^{4})$ if $Q$ satisfies some conditions. As an application, we consider the divisor problem with respect to the coefficients of the Dirichlet series of Epstein zeta functions. Certain estimates for the error term of the sum of the Dirichlet coefficients are obtained by combining our results and Fomenko's estimates for $\zeta (\frac{n-1}{2}+it;Q)$.

#### Article information

Source
Tokyo J. Math., Volume 36, Number 1 (2013), 269-287.

Dates
First available in Project Euclid: 22 July 2013

https://projecteuclid.org/euclid.tjm/1374497524

Digital Object Identifier
doi:10.3836/tjm/1374497524

Mathematical Reviews number (MathSciNet)
MR3112388

Zentralblatt MATH identifier
1355.11027

#### Citation

SONO, Keiju. On the Fourth Moment of the Epstein Zeta Functions and the Related Divisor Problem. Tokyo J. Math. 36 (2013), no. 1, 269--287. doi:10.3836/tjm/1374497524. https://projecteuclid.org/euclid.tjm/1374497524

#### References

• O. M. Fomenko, Order of the Epstein zeta-function in the critical strip, J. of Math. Sci. 110 (2002), No. 6, 3150–3163.
• G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41 (1918), 119–196.
• D. R. Heath-Brown, Hybrid bounds for Dirichlet $L$-functions I, Invent. Math. 47 (1978), 149–170.
• D. R. Heath-Brown, Hybrid bounds for Dirichlet $L$-functions II, Quart. J. Math. 31 (1980), 157–167.
• D. R. Heath-Brown, Fractional moments of Dirichlet $L$-functions, Acta Arith. 145 (2010), No. 4, 397–409.
• D. R. Heath-Brown, The twelfth power moment of the Riemann zeta-function, Quart. J. Math. Oxford Ser. (2) 29 (1978), 443–462.
• E. Hecke, Über Modulfunktionen und Dirichletschen Reihen mit Eulerscher Productentwicklung I, II, Math. Ann. 114 (1937), 1–28, 316–351.
• A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. 27 (1926), 273–300.
• H. Iwaniec, Topics in Classical Automorphic Forms, Amer. Math. Soc. Graduate Studies in Mathematics
• S. Kanemitsu, A. Sankaranarayanan and Y. Tanigawa, A Mean Value Theorem for Dirichlet Series and a General Divisor Problem, Monatsh. Math. 136 (2002), 17–34.
• G. Lü, On a divisor problem related to the Epstein zeta-function, Bull. London Math. Soc. 42 (2010), 267–274.
• G. Lü, J. Wu and W. Zhai, On a divisor problem related to the Epstein zeta-function II, J. of Number Theory 131 (2011), 1734–1742.
• A. V. Malyshev, Representation of integers by positive quadratic forms (in Russian), Trudy Mat. Inst. Akad. Nauk SSSR, 65 (1962).
• H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Math. $\mathbf{227}$, Springer, Berlin, 1971.
• W. Müller, The mean square of Dirichlet series associated with automorphic forms, Mh. Math. 113 (1992), 121–159.
• A. Sankaranarayanan, On a divisor problem related to the Epstein zeta-function, Arch. Math. 65 (1995), 303–309.
• C. L. Siegel, Contribution to the theory of the Dirichlet $L$-series and the Epstein zeta-functions, Ann. of Math. 44 (1943), 143–172.
• E. C. Titchmarsh, The theory of the Riemann zeta-function, Oxford, Calendon Press, 1951.