Bounds for double zeta-functions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 445-464.

In this paper we shall derive the order of magnitude for the double zeta-functionof Euler-Zagier type in the region 0s j <1(j=1,2).First we prepare the Euler-Maclaurinsummation formula in a suitable form for our purpose, and then we apply the theory of doubleexponential sums of van der Corput’s type.

Classification : 11L07, 11M41
@article{ASNSP_2006_5_5_4_445_0,
     author = {Kiuchi, Isao and Tanigawa, Yoshio},
     title = {Bounds for double zeta-functions},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {445--464},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {4},
     year = {2006},
     mrnumber = {2297719},
     zbl = {1170.11317},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_4_445_0/}
}
TY  - JOUR
AU  - Kiuchi, Isao
AU  - Tanigawa, Yoshio
TI  - Bounds for double zeta-functions
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2006
SP  - 445
EP  - 464
VL  - 5
IS  - 4
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2006_5_5_4_445_0/
LA  - en
ID  - ASNSP_2006_5_5_4_445_0
ER  - 
%0 Journal Article
%A Kiuchi, Isao
%A Tanigawa, Yoshio
%T Bounds for double zeta-functions
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2006
%P 445-464
%V 5
%N 4
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2006_5_5_4_445_0/
%G en
%F ASNSP_2006_5_5_4_445_0
Kiuchi, Isao; Tanigawa, Yoshio. Bounds for double zeta-functions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 445-464. http://www.numdam.org/item/ASNSP_2006_5_5_4_445_0/

[1] S. Akiyama, S. Egami and Y. Tanigawa, An analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith. 98 (2001), 107-116. | MR | Zbl

[2] S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J. 5 (2001), 327-351. | MR | Zbl

[3] F. V. Atkinson, The mean-value of the Riemann zeta-function, Acta Math. 81 (1949), 353-376. | MR | Zbl

[4] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Higher Transcendental Functions I”, McGraw-Hill, New-York, 1953. | Zbl

[5] I. S. Gradshteyn and I. M. Ryzhik, “Tables of Integrals, Series and Products”, Academic Press, Inc., 1979. | Zbl

[6] S. W. Graham and G. Kolesnik, “Van der Corput's methods of exponential sums”, London Math. Society, LNS Vol. 126, Cambridge University Press 1991. | MR | Zbl

[7] J. L. H. Hafner, New omega results in a weighted divisor problem, J. Number Theory 28 (1988), 240-257. | MR | Zbl

[8] M. N. Huxley, “Area, Lattice Points and Exponential Sums”, Oxford Science Publications, Clarendon Press, Oxford, 1996. | MR | Zbl

[9] M. N. Huxley, Integer points, exponential sums and the Riemann zeta function, In: “Number Theory for the Millennium, Proc. Millennial Conf. Number Theory", Vol. II, M. A. Bennett et al. (eds.), A K Peters 2002, 275-290. | MR | Zbl

[10] H. Ishikawa and K. Matsumoto, On the estimation of the order of Euler-Zagier multiple zeta-functions, Illinois J. Math. 47 (2003), 1151-1166. | MR | Zbl

[11] A. Ivić, “The Riemann Zeta-Function”, John Wiley & Sons, 1985. | MR | Zbl

[12] E. Krätzel, “Lattice Points”, Kluwer Academic Publishers, 1988. | MR | Zbl

[13] K. Matsumoto, On the analytic continuation of various multiple zeta-functions, In: “Number Theory for the Millennium, Proc. Millennial Conf. Number Theory”, Vol. II, M. A. Bennett et al. (eds.), A K Peters 2002, 417-440. | MR | Zbl

[14] Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory 74 (1999), 39-43. | MR | Zbl

[15] E. C. Titchmarsh, On Epstein's zeta-function, Proc. London Math. Soc. (2) 36 (1934), 485-500. | Zbl

[16] E. C. Titchmarsh, “The Theory of the Riemann Zeta-Function” (Revised by D. R. Heath-Brown), Clarendon press Oxford, 1986. | MR | Zbl

[17] J. Q. Zhao, Analytic continuation of multiple zeta function, Proc. Amer. Math. Soc. 128 (2000), 1275-1283. | MR | Zbl