On the Error Term for the Mean Value Associated With Dedekind Zeta-function of a Non-normal Cubic Field

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1 Λ44ΩΛ6fi Ψ ο Vol.44, No.6 05ffμ ADVANCES IN MAHEMAICS(CHINA) Nov., 05 doi: 0.845/sxjz.04038b On the Error erm for the Mean Value Associated With Dedekind Zeta-function of a Non-normal Cubic Field SHI Sanying (School of Mathematics, Hefei University of echnology, Hefei, Anhui, 30009, P. R. China) Abstract: Let E 3/Q be a non-normal cubic extension field, and let a k be the number of integral ideals in E 3 with norm k. Denote R(x) by the remainder term in the asymptotic formula for the average behavior a k.inthispaper,itisshownthat R (x)dx ε X ε for any given ε>0. Keywords: Dedekind zeta-function; number fields; mean value MR(00) Subject Classification: N37; R4 / CLC number: O56.4 Document code: A Article ID: (05) Introduction and Main Results Let E be an algebraic number field of degree d over the rational field Q, andζ(s, E) beits Dedekind zeta-function. hus for Re(s) >, ζ(s, E) = a (Na) s, where a runs over all integral ideals of the field E, andna is the norm of a. Leta k denote the number of integral ideals in E with norm k. henwehave ζ(s, E) = k= a k, s = σ +it, σ >. ks It is known that a k is a multiplicative function and satisfies a k τ(k) d, () where τ(k) is the divisor function. Received date: Revised date: Foundation item: Supported by NSFC (No. 007, No. 0786) and the Natural Science Foundation of Anhui Province (No QA0). vera3 99@hotmail.com

2 846 ffl Φ ν 44Ω It is a classical problem to study behaviors of the function a k. In 97, Landau [0] first proved that a k = αx + O ( x +ε) d+ for any arbitrary algebraic number field of degree d, where α is the residue of ζ(s, E) at its simple pole s =. Later, Huxley and Watt [6] and Müller [] improved the results for the quadratic and cubic fields, respectively. For any arbitrary algebraic number field of degree d 3, Nowak [3] obtained the best result O ( x d ) d(5d+) (log x) 5d+ for 3 d 6, a n = αx + O ( x d + ) 3 d (log x) d for d 7. Chandrasekharan and Narasimhan [] first studied the mean square value of a k.in[],they showed that if E is a Galois extension of Q of degree d, then a k cx logd x for a suitable constant c = c(e). In the same paper, they also considered the non-normal extension field, and obtained the upper bound for the mean square of a k, a k x log d x, x. () In 008, Fomenko [] considered a non-normal cubic extension E 3 /Q. Based on the fact that the strong Artin conjecture holds true for the Galois group, he refined the result (), and established the asymptotic formula: a k = b x log x + b x + O ( x 9 +ε) (3) for the field K 3,whereb,b are constants. Recently, the exponent 9 in the error term for the asymptotic formula (3) was improved to 3 3 by Lü[]. he aim of this paper is to study the continuous mean square of the error term in the asymptotic formula (3), and we prove the following result. heorem. Let E 3 /Q be a non-normal cubic extension field, and define R(x) := a k (b x log x + b x). (4) hen we have for any given ε>0. R (x)dx ε X ε

3 6fi ffifl»: Mean Value Associated With Dedekind Zeta-function 847 Notations As usual, the Vinogradov symbol A B means that B is positive and the ratio A/B is bounded. he letter ε denotes an arbitrary small positive number, not the same at each occurrence. Proof of heorem. o prove our theorem, we need the following lemmas. Lemma. Let L(s) bedefinedin(7)andf be fixed by an irreducible polynomial f(x) = x 3 + ax + bx + c. LetL(s, f) andl(s, sym f)denotetheheckel-function and the symmetric square L-function of cusp form f, respectively. hen for Re(s) >, we have L(s) =ζ (s)l (s, f)l(s, sym f)a(s), where A(s) is a Dirichlet series, which converges uniformly and absolutely in the half plane Re(s) > + ε for any ε>0. Proof his is [, Lemma.4]. Lemma. For any ε>0, we have the subconvexity bound ( ) L +it, f f,ε ( t +) max{ 3 ( σ),0}+ε (5) uniformly for σ and t, and ( ) 6 L +it, f dt +ε (6) uniformly for. Proof he result (5) is due to Good [4], and (6) was established by Jutila [9]. Lemma.3 For any ε>0, we have L(σ +it, sym f) f,ε t max{ 8 ( σ),0}+ε uniformly for σ and t. Proof his is the result () of [, Lemma.3]. Lemma.4 We have ( ) ζ +it dt log 7. Proof his lemma is heorem in Heath-Brown [5]. Now we begin to prove our theorem. Let us define L(s) = a k k s (7) for Re(s) >. From Lemma. and the arguments in Gelbart and Jacquet [3], we see that k= L(s) =ζ (s)l (s, f)l(s, sym f)a(s)

4 848 ffl Φ ν 44Ω has an analytic continuation to the right of the line Re(s) =. Let = X 8 7. From (), (7) and Perron s formula (see [8, Proposition 5.54]), we get a k = +ε+i πi +ε i s ds + O ( x +ε By the property L(s) only has a simple pole at s =forre(s) > and Cauchy s residue theorem, we have where { a k = +ε+i +ε+i + + πi +ε i +ε+i +ε i +ε i } s := x(b log x + b )+J (x)+j (x)+j 3 (x)+o(x +ε ), J (x) := πi J (x) := πi J 3 (x) := πi From the definition of R(x) in(4),wehave +ε+i +ε+i +ε i s ds, +ε+i s ds, +ε i +ε i s ds. ). R(x) =J (x)+j (x)+j 3 (x)+o(x +ε ). herefore to prove heorem., we shall prove the following results: I i = ds + Res s= L(s)x + O ( ) x +ε J i (x)dx ε X ε, i =,, 3 (8) and It is easy to get ( O ( x +ε ( ( )) x +ε ( ) X 3+ε O dx = O )) dx ε X ε. (9) X ε. (0) Now we consider the integral J (x). We have J (x) = ( ) x L π + ε +it +ε+it dt. + ε +it

5 6fi ffifl»: Mean Value Associated With Dedekind Zeta-function 849 hen I = J (x)dx = ( ( 4π L + ε +it ( ) x L + ε +it +ε it + ε it ) x +ε+it + ε +it dt )dx dt ( x )dxdt +ε+i(t t) dt = L( + ε +it )L( + ε +it ) 4π ( + ε +it )( + ε it ) X +ε L( dt + ε +it ) L( + ε +it ) ( + t )( + t )( + t t ) dt ( L( X +ε dt + ε +it ) ( + t ) + L( + ε +it ) ) dt ( + t ) + t t X +ε L( + ε +it ) dt ( + t ) dt + t t. () o go further, we get dt + t t t+ + t t + + t log. ( dt + dt t t dt t + t + t ) dt t t () By () and (), I X +ε L( log + ε +it ) ( + t ) dt. (3) From (3) and Lemma., ( ) I X +3ε + X +3ε ζ + ε +it ( ) ( ) L + ε +it, f L + ε +it, sym f t dt { ( X +3ε + X +3ε max 8 ( ) ) ζ + ε +it, E 3 dt (4) ( ( ) 6 ) } L + ε +it, f 3 dt X +3ε + X +4ε 8 X ε,

6 850 ffl Φ ν 44Ω wherewehaveusedlemma.3, and ( ) ζ + ε +it dt log 7, ( ) 6 L + ε +it, f dt +ε, which can be deduced from Lemma.4 and (6) and Gabriel s convexity theorem, see e.g. [7, Lemma 8.3] respectively. Finally, we estimate trivial bounds of the integrals J (x) andj 3 (x). Applying (5), Lemma.3 and the well-known bound for Riemann zeta function we get ζ(σ +it) ( t +) σ 3 +ε, which yields J (x)+j 3 (x) +ε x σ ζ (σ +i )L (σ +i,f)l(σ +i,sym f) dσ +ε max x σ ( σ) +ε σ +ε ( x = max +ε σ +ε 7 8 x+ε + x +ε 6 +ε, I + I 3 +ε 4( σ) 3 +ε ( σ) 8 +ε ) σ 7 8 +ε (J (x)+j 3 (x)) dx ( ) x +ε + x +ε 6 +ε dx ) dx + ( x +ε X3+ε + X +3ε 8 X ε. ( x +ε 6 +ε) dx he inequalities (8) (9) immediately follow from (0), (4) and (5). hat is, (5) hen this completes the proof of heorem.. References R (x)dx ε X ε. [] Chandrasekharan, K. and Narasimhan, R., he approximate functional equation for a class of zeta-functions, Math. Ann., 963, 5():

7 6fi ffifl»: Mean Value Associated With Dedekind Zeta-function 85 [] Fomenko, O.M., Mean values connected with the Dedekind zeta function, J. Math. Sci. (N. Y.), 008, 50(3): 5-. [3] Gelbart, S. and Jacquet, H., A relation between automorphic representations of GL() and GL(3), Ann. Sci. École Norm. Sup., 978, (4): [4] Good, A., he square mean of Dirichlet series associated with cusp forms, Mathematika, 98, 9(): [5] Heath-Brown, D.R., he twelfth power moment of the Riemann-function, Q. J. Math., 978, 9(4): [6] Huxley, M.N. and Watt, N., he number of ideals in a quadratic field II, Israel J. Math. Part A, 000, 0: [7] Ivić, A., Exponent pairs and the zeta function of Riemann, Studia Sci. Math. Hungar., 980, 5(//3): [8] Iwaniec, H. and Kowalski, E., Analytic number theory, Amer. Math. Soc. Colloq. Publ., Vol. 53, Providence, RI: AMS, 004, [9] Jutila, M., Lectures on A Method in the heory of Exponential Sums, ata Inst. Fund. Res. Stud. Math., Berlin: Springer-Verlag, 987. [0] Landau, E., Einführung in die elementare and analytische heorie der algebraischen Zahlen und der Ideale, Berlin: eubner, 97 (in German). [] Lü, G.S., Mean values connected with the Dedekind zeta-function of a non-normal cubic field, Cent. Eur. J. Math., 03, (): [] Müller, W., On the distribution of ideals in cubic number fields, Monatsh. Math., 988, 06(3): -9. [3] Nowak, W.G., On the distribution of integral ideals in algebraic number fields, Math. Nachr., 993, 6(): ψ+ffi0.%ρ-flfffi# ζ!&fl$fl,)(' 34 (±ΞΠ ffl, ±Ξ,, 30009) /* M E 3 /Q PV>= Z:K7FX, a k 6OYX E 3 L<QR k :[G:>Q. R x 6OAN a k :BDN:WU. 5S]IH;J?: ε>0, χ"ß 89C QX; E^ R (x)dx ε X ε.