On the mean square of the error term for Dedekind zeta functions

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 30, Tome 440 (2015), pp. 187-204

Résumé

Let $K_n$ be a number field of degree $n$ over $\mathbb Q$. Denote by $D(x,K_n)$ the number of all non-zero integral ideals in $K_n$ with norm $\leq x$. The Dedekind zeta function $\zeta_{K_n}(s)$ is a meromorphic function with a simple pole at $s=1$, with residue, say, $\Lambda_n$. Define $$ \Delta(x, K_n)=D(x, K_n)-\Lambda_n x. $$ The history of estimates of $\Delta(x,K_n)$ begins with $$ \Delta (x, K_n)\ll x^{1-\frac1n}\qquad\text{(Weber (1896))} $$ and $$\Delta(x, K_n)\ll x^{\frac{n-1}{n+1}}\qquad\text{(Landau (1917))}. $$ If $n>2$, then $$ \int^x_1\Delta(y, K_n)^2\,dy\ll x^{3-\frac4n}\log^nx, $$ which is a result of Chandrasekharan and Narasimhan (1964). In this paper the following new results are obtained. 1) For $K_4=\mathbb Q(\root4\of{m})$, $m>1$ is square-free, the author proves $$ x^{\frac74}\ll\int^x_1\Delta(y,K_4)^2dy\ll x^{\frac74+\varepsilon}. $$ 2) For $K_6$, the normal closure of a cubic field $K_3$ with the Galois group $S_3$ and discriminant $\Delta0$, the author proves $$ x^{\frac{11}6}\ll\int^x_1\Delta(y,K_6)^2\,dy\ll x^{2+\varepsilon}. $$