Journal of Mathematical Sciences

, Volume 234, Issue 5, pp 737–749

# On Riesz Means of the Coefficients of Epstein’s Zeta Functions

• O. M. Fomenko
Article
Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$\sqrt{n}$$. The generating function
$${\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2,$$
is Epstein’s zeta function. The paper considers the Riesz mean of the type
$${D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n),$$
where ρ > 0; the error term Δρ(x; ζ3) is defined by
$${D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right).$$
K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that
$${\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{\begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}}$$
In the present paper, it is proved that
$${\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{\begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2<\rho <1\right),\\ {}O\left({x}^{3/4+\rho /4+\varepsilon}\right)& \left(0<\rho \le 1/2\right),\end{array}}$$

and the Riesz means of the coefficients of ζk(s), k ≥ 4, are studied.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
D. R. Heath-Brown, “Lattice points in the sphere,” in: Number Theory in Progress, Vol. 2, de Gruyter, Berlin (1999), pp. 883–892.Google Scholar
2. 2.
Y.-K. Lau and K.-M. Tsang, “Large values of error terms of a class of arithmetical functions,” J. reine angew. Math., 544, 25–38 (2002).
3. 3.
O. M. Fomenko, “Lattice points in the circle and the sphere,” Zap. Nauchn. Semin. POMI, 418, 198–220 (2013).Google Scholar
4. 4.
K. Chandrasekharan and R. Narasimhan, “Functional equations with multiple gamma factors and the average order of arithmetical functions,” Ann. Math., 76, 93–136 (1962).
5. 5.
F. Chamizo, “Lattice points in bodies of revolution,” Acta Arithm., 85, 265–277 (1998).
6. 6.
K. Chandasekharan and R. Narasimhan, “Hecke’s functional equation and the average order of arithmetical functions,” Acta Arithm., 6, 487–503 (1961).
7. 7.
S. Krupička, “On the number of lattice points in multidimensional convex bodies,” Czech. Math. J., 7, 524–552 (1957).
8. 8.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Second edition revised by D. R. Heath-Brown, Clarendon Press, Oxford (1986).Google Scholar
9. 9.
K. Prachar, Primzahlverteilung, Berlin (1957).Google Scholar
10. 10.
E. Landau, Ausgewählte Abhandlungen zur Gitterpunktlehre, Berlin (1962).Google Scholar
11. 11.
F. Chamizo and H. Iwaniec, “On the sphere problem,” Rev. Mat. Iberoamer., 11, 417–429 (1995).
12. 12.
Y.-K. Lau, “On the mean square formula for the error term for a class of arithmetical functions,” Mh. Math., 128, 111–129 (1999).
13. 13.
K.-C. Tong, “On divisor problems. I, III,” Acta Math. Sinica, 5, 313–324 (1955); 6, 515–541 (1956).Google Scholar
14. 14.
O. M. Fomenko, “On the mean square of the error term for the Dedekind zeta functions,” Zap. Nauchn. Semin. POMI, 440, 187–204 (2015).Google Scholar
15. 15.
J. L. Hafner, “On the representation of the summatory functions of a class of arithmetical functions,” Lect. Notes Math., 899, 148–165 (1981).
16. 16.
D. R. Heath-Brown, “The distribution and moments of the error term in the Dirichlet divisor problem,” Acta Arithm., 60, 389–415 (1992).
17. 17.
A. Walfisz, Weylsche Exponentialsummen in der Neueren Zahlentheorie, Berlin (1963).Google Scholar
18. 18.
O. M. Fomenko, “Lattice points in many-dimensional balls,” Zap. Nauchn. Semin. POMI, 449, 261–274 (2016).Google Scholar
19. 19.
K. Ramachandra and A. Sankaranarayanan, “Hardy’s theorem for zeta functions of quadratic forms,” Proc. Indian Acad. Sci. (Math. Sci.), 106, 217–226 (1996).