The signs of the Stieltjes constants associated with the Dedekind zeta function
aa r X i v : . [ m a t h . N T ] D ec THE SIGNS OF THE STIELTJES CONSTANTS ASSOCIATED WITH THEDEDEKIND ZETA FUNCTION
SUMAIA SAAD EDDIN
Abstract.
The Stieltjes constants γ n ( K ) of a number field K are the coefficients of theLaurent expansion of the Dedekind zeta function ζ K ( s ) at its pole s = 1. In this paper, weestablish a similar expression of γ n ( K ) as Stieltjes obtained in 1885 for γ n ( Q ). We also studythe signs of γ n ( K ). Introduction
For a number field K , the Dedekind zeta function is defined ζ K ( s ) = X a N a s = Y p − N p − s , ℜ ( s ) > . Here, a runs over non-zero ideals in O K , the ring of integers of K , p runs over the primeideals in O K and N a is the norm of a . It is known that ζ K ( s ) can be analytically continuedto C − { } , and that at s = 1 it has a simple pole, with residue γ − ( K ) given by γ − ( K ) = 2 r (2 π ) r h ( K ) R ( K ) ω ( K ) p | d ( K ) | , where r denotes the number of real embeddings of K , r is the number of complex embeddingsof K , h ( K ) is the class number of K , R ( K ) is the regulator of K , ω ( K ) is the number ofroots of unity contained in K and d ( K ) is the discriminant of the extension K/ Q . Further,the Laurent expansion of ζ K ( s ) at s = 1 is(1) ζ K ( s ) = γ − ( K ) s − X n ≥ γ n ( K )( s − n . The coefficients γ n ( K ) are sometimes called the Stieltjes constants associated with the Dedekindzeta function. In [6], they are called by higher Euler’s constants of K . While the constant γ K = γ ( K ) /γ − ( K ) is called the Euler-Kronecker constant in Ihara [7] and Tsfasman [16].In case K = Q , the Laurent expansion of the Riemann zeta function ζ ( s ) at its pole s = 1is given by ζ ( s ) = 1 s − X n ≥ γ n ( s − n , where(2) γ n = ( − n n ! lim x →∞ x X m =1 (log m ) n m − (log x ) n +1 ( n + 1) ! . Date : 06 July, 2017.
Mathematics Subject Classification (2000) . 11R42; 11M06
Stieltjes in 1885 was the first to propose this definition of γ n , for this reason these constantsare called today by his name. The asymptotic behaviour of γ n , as n → ∞ has been widelystudied by many authors ( for instance: Briggs [3], Mitrovi`c [12] , Israilov [8], Matsuoka [11]and more recently Coffey [4] and [5], Knessl and Coffey [9], Adell [2], Adell and Lekuona [1]and Saad Eddin [14]). Their main interest is focused on the growth, the sign changes ofthe sequence ( γ n ) and on giving explicit upper estimates for | γ n | . Moreover, they obtainedrelations between this sequence and the zeros of ζ ( s ) ( see [11], [15]).In this paper we are interested in the Stieltjes coefficients γ n ( K ) for the Dedekind zetafunction. We first give the following formula of γ n ( K ) which is similar to Stieltjes’s formulagiven by Eq (2). Theorem 1.
For any n ≥ , we have γ n ( K ) = ( − n n ! lim x →∞ X N a ≤ x (log N a ) n N a − γ − ( K ) (log x ) n +1 n + 1 , and γ ( K ) = lim x →∞ X N a ≤ x N a − γ − ( K ) log x + γ − ( K ) . This result seems similar to another result obtained by Hashimoto et al [6] for the higherEuler-Selberg constants. Despite a considerable effort the author have not been able to findTheorem 1 in the literature.In 1962, Mitrovi`c [12] studied the sign changes of the constants γ n and prove that; Each ofthe inequalities γ n > , γ n < , γ n − > , γ n − < , holds for infinitely many n . In [11], Matsuoka gave precise conditions for the sign of γ n . Bythe same techniques used in [12], we prove that Theorem 2.
For the coefficients in the expansion (1) , each of the inequalities γ n ( K ) > , γ n ( K ) < , γ n − ( K ) > , γ n − ( K ) < , holds for infinitely many n . It immediately follows that
Corollary 1.
Infinitely many γ n ( K ) are positive and infinitely many are negative. Proofs
Proof of Theorem 1.
By Eq (1), we note that ζ K ( s ) − γ − ( K ) ss − ζ K ( s ) − γ − ( K ) s − − γ − ( K )= X n ≥ α n ( K )( s − n , (3)where α ( K ) = γ ( K ) − γ − ( K ) and α n ( K ) = γ n ( K ) for n ≥
1. By the definition of ζ K ( s ),we write ζ K ( s ) = Z + ∞ − dN K ( t ) t s = s Z + ∞ − N K ( t ) t s +1 dt, HE SIGNS OF THE STIELTJES CONSTANTS ASSOCIATED WITH THE DEDEKIND ZETA FUNCTION 3 where N K ( t ) = X N a ≤ t . Then, we get(4) ζ K ( s ) − γ − ( K ) ss − s Z + ∞ − N K ( t ) − γ − ( K ) tt s +1 dt. Put P n ≥ α n ( K )( s − n = h ( s ). From Eq(3) and Eq (4), we have h ( s ) = s Z + ∞ − N K ( t ) − γ − ( K ) tt s +1 dt. From [10, Satz 210], we have N K ( t ) = γ − ( K ) t + O (cid:0) t − /m (cid:1) , where m is the degree of K and Q . For ℜ s > − /m , it is easily seen that the n -th derivative of h ( s ) at s = 1 is(5) h ( n ) (1) = n ! α n ( K ) = ( − n ( I − I ) , where I = Z + ∞ − N K ( t ) (cid:18) log n t − n (log t ) n − t (cid:19) dt, and I = γ − ( K ) Z + ∞ − log n t − n (log t ) n − t dt. On the other hand, we have X N a ≤ x (log N a ) n N a = Z x − log n tt dN K ( t ) = N K ( x ) log n xx + Z x − N K ( t ) (cid:18) log n t − n (log t ) n − t (cid:19) dt. Thus, we get Z x − N K ( t ) (cid:18) log n t − n (log t ) n − t (cid:19) dt = X N a ≤ x (log N a ) n N a − N K ( x ) log n xx . Again using the fact that N K ( t ) = γ − ( K ) t + O (cid:0) t − /m (cid:1) , we find Z x − N K ( t ) (cid:18) log n t − n (log t ) n − t (cid:19) dt = X N a ≤ x (log N a ) n N a − γ − ( K ) log n x + O (cid:18) log n xx /m (cid:19) . Taking x → + ∞ , the above becomes(6) I = lim x → + ∞ X N a ≤ x (log N a ) n N a − γ − ( K ) log n x . Now, notice that(7) I = lim x → + ∞ (cid:20) γ − ( K ) (log x ) n +1 n + 1 − γ − ( K ) log n x (cid:21) From Eq (5) and (6) and(7), we conclude that, for n ≥ γ n ( K ) = α n ( K ) = ( − n n ! lim x →∞ X N a ≤ x (log N a ) n N a − γ − ( K ) (log x ) n +1 n + 1 and γ ( K ) = α ( K ) + γ − ( K ). This completes the proof. (cid:3) SUMAIA SAAD EDDIN
Proof of Theorem 2.
To prove Theorem 2, we apply the same technique used in [12]. Let C be the set of all positive integers n such that γ n ( K ) = 0. Define C = { n : γ n ( K ) = 0 and ( − n = 1 } C − = { n : γ n ( K ) < − n = 1 } ,C +1 = { n : γ n ( K ) > − n = 1 } , and C = { n : γ n ( K ) = 0 and ( − n = − } ,C − = { n : γ n ( K ) < − n = − } ,C +2 = { n : γ n ( K ) > − n = − } . From [13], we have ζ K ( s ) − γ − ( K ) s − transcendental function. So the cardinal number of the set C is equal to thecardinal number of the set of all positive integers ℵ . Then, we can write ζ K ( s ) − γ − ( K ) s − (cid:16) X n ∈ C − + X n ∈ C +1 + X n ∈ C − + X n ∈ C +2 (cid:17) γ n ( K )( s − n . Replacing s by t + 1 and then by − t + 1 in the above. Adding and then subtracting the results,we find that(8) ζ K ( t + 1) + ζ K ( − t + 1) = 2 (cid:16) X n ∈ C − + X n ∈ C +1 (cid:17) γ n ( K ) t n , and(9) ζ K ( t + 1) − ζ K ( − t + 1) − γ − ( K ) t = 2 (cid:16) X n ∈ C − + X n ∈ C +2 (cid:17) γ n ( K ) t n . Taking t = 2 m + 1 with m > ζ K ( s ) vanishes at all negative evenintegers. We find the left hand side of Eq (8) approaches to 1 when m → + ∞ . It follows thatthe right hand side of this equation can’t be polynomial. Therefore the cardinal of the set C is ℵ . On the other hand, if we assume that the cardinal of the set C − is less than ℵ . Thenthe right hand side of Eq (8) approaches + ∞ . Similarly, if the cardinal of the set C +1 is lessthan ℵ . Then the right hand side of Eq (8) approaches −∞ , this leads to a contradiction.We thus conclude that the cardinal of the sets C − and C +1 are ℵ . By a similar argument,we show that the cardinal of the sets C − and C +2 are ℵ . That completes the proof. (cid:3) Acknowledgement
The author would like to thank Professor Kohji Matsumoto for his valuable commentson an earlier version of this paper. The author is supported by the Japan Society for thePromotion of Science (JSPS) “ Overseas researcher under Postdoctoral Fellowship of JSPS”.Part of this work was done while the author was supported by the Austrian Science Fund(FWF) : Project F5507-N26, which is part of the special Research Program “ Quasi MonteCarlo Methods : Theory and Application”.
HE SIGNS OF THE STIELTJES CONSTANTS ASSOCIATED WITH THE DEDEKIND ZETA FUNCTION 5
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