arXiv:1003.4813v3 [math.GM] 16 Jan 2017 O where ovretDrcltseries Dirichlet convergent utetring quotient Let Introduction 1 classification: subject AMS Keywords: Abstract. eafnto of function zeta K of K eeidzt ucinand function zeta Dedekind K a ea leri ubrfil ihdegree with field number algebraic an be agstruhtennzr daso h leri neesring integers algebraic the of ideals non-zero the through ranges and ita cdm fSineadTechnology and Science of Academy Vietnam 8HagQo it 00 ao,Vietnam Hanoi, 10307 Viet, Quoc Hoang 18 O | a K | K / eoe h bouenr of norm absolute the denotes a -al [email protected] e-mail: ti elkonthat known well is It . ao nttt fMathematics of Institute Hanoi D conjecture BDS sfis endfrcmlxnumbers complex for defined first is oebr1,2018 13, November agV Giang Vu Dang ζ K ( 14R15 s = ) 1 X a | | a ab | − | s = a | a hc stecriaiyof cardinality the is which || n b | [ = . o vr ideals every for s K with : Q .Dedekind ]. ℜ ( s ) > a by 1 and b of OK. Clearly, if K = Q, the Dedekind zeta function ζQ of Q becomes the classical ζ. Moreover, for K = Q(e2πi/m),

−s −1 ζ 2πi (s)= ζ(s) L(s, χ) (1 −|p| ) . Q(exp m ) Y Y χ p|m Here, χ denotes a non-principal modulo m. The of Dedekind zeta function of K is a product over all the prime ideals p of OK

1 µ (a) p −s K = (1 −| | )= s , for Re(s) > 1. ζK(s) Y X |a| p a This is the expression in analytic terms of uniqueness of the prime factoriza- tion of an a in OK (). The M¨obius function µK in the field K is defined by

1 if a = OK  r µK(a)= (−1) if a = p1p2 ··· pr . 0 if p2|a  For example, µK(5) = 1 for K = Q(i), but µQ(5) = µ(5) = −1. For ℜ(s) > 1, ζK(s) is given by a convergent product of infinite non-zero numbers, so ζK(s) is non-zero in this region. Erich Hecke first proved that ζK(s) has an to the as a , having a simple pole only at s =1. Let ∆K denote the discriminant of K, r1 and r2 denote the number of real and complex places of K, and

−s/2 −s ΓR(s)= π Γ(s/2), ΓC(s) = 2(2π) Γ(s).

s/2 r1 r2 1 2 1 1 ΛK(s)= |∆K| ΓR(s) ΓC(s) ζK(s) ΞK(s)= 2 (s + 4 )ΛK( 2 + is). We have the functional equation

ΛK(s)=ΛK(1 − s) ΞK(−s)=ΞK(s).

Moreover, 2r1+r2 πr2 h(K)R(K) lim(s − 1)ζK(s)= . s→1 w(K) |∆K| p In the virtue of this functional equation, it is also conjectured that the com- plex roots of ζK(s) are on the critical line ℜs = 1/2. This is called the

2 extended . If this hypothesis is true then every complex root of Riemann zeta function ζ and Dirichlet L−function L(χ) is on the line ℜs =1/2. The values of the Dedekind zeta function at integers encode important arithmetic data of the field K. For example, the class number h(K) of K relates the residue at s = 1, the regulator R(K) of K, the number w(K) of unity roots in K, the discriminant of K, and the number of real and complex places of K. From the functional equation and the fact that Γ is infinite at all integers less than or equal to zero, it follows that ζK(s) vanishes at all negative even integers. It even vanishes at all negative odd integers unless K is totally real number field. In the totally real case, Carl Ludwig Siegel showed that ζK(s) is a non-zero rational number at negative odd in- tegers. Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K−theory of K. Another example is at s = 0 where ζK(s) has a zero with order r equal to the rank the group of OK and the leading term given by

−r h(K)R(K) lim s ζK(s)= − . s→0 w(K) Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. There are examples of non-isomorphic fields that are arith- metically equivalent. In particular some of these fields have different class numbers, so the Dedekind zeta function of a number field does not determine its class number. Now note that

∞ 1 ln ζK(s)= , X X n|p|ns n=1 p

∞ pn +1 ln ζ(s) + ln ζ(s − 1) = . X X n|p|ns n=1 p For an elliptic curve E over Q we define

∞ #E(F n ) ln L(E,s)= p X X n|p|ns n=1 p6| 2N

n 2 We have #E(Fpn )= p +1 if E denotes the elliptic curves y = x(x−N)(x+ N) and p =4k + 3.

3 Acknowledgement. Deepest appreciation is extended towards the NAFOSTED (the National Foundation for Science and Techology Development in Vietnam) for the financial support.

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