Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T08:49:29.106Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Conjecture of Chowla and Milnor

Published online by Cambridge University Press:  20 November 2018

Sanoli Gun ,
M. Ram Murty  and
Purusottam Rath
Sanoli Gun
Affiliation:
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India email: [email protected]
M. Ram Murty
Affiliation:
Department of Mathematics, Queen’s University, Kingston, ON K7L 3N6, Canada email: [email protected]
Purusottam Rath
Affiliation:
Chennai Mathematical Institute, Plot No H1, SIPCOT IT Park, Padur PO, Siruseri 603103, Tamil Nadu, India email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of $L$-functions associated to periodic functions at integers greater than 1. We report on some progress in relation to these conjectures. In a different vein, we link them to a conjecture of Zagier on multiple zeta values and also to linear independence of polylogarithms.

Keywords

11F2011F11
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 6 , 01 December 2011 , pp. 1328 - 1344
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Baker, A., Birch, B. J. and E. A.Wirsing, On a problem of Chowla. J. Number Theory 5(1973), 224–236. doi:10.1016/0022-314X(73)90048-6Google Scholar
[2] Ball, K. and Rivoal, T., Irrationalité d’une infinité de valeurs de la fonction zeta aux entiers impairs. Invent. Math. 146(2001), 193–2007. doi:10.1007/s002220100168Google Scholar
[3] Chowla, S., The nonexistence of nontrivial linear relations between the roots of a certain irreducible equation. J. Number Theory 2(1970), 120–123. doi:10.1016/0022-314X(70)90012-0Google Scholar
[4] Chowla, P. and Chowla, S., On irrational numbers. Norske Vid. Selsk. Skr. (Trondheim) 3(1982), 1–5. (See also Chowla, S., Collected Papers, Vol. 3. CRM, Montreal, 1999, 1383–1387.)Google Scholar
[5] Goncharov, A., Multiple A-values, Galois groups, and geometry of modular varieties. In: European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, Basel, 2001, 361–392.Google Scholar
[6] Kumar Murty, V. and Ram Murty, M. , Transcendental values of class group L-functions. Math Annalen, Published online, December 2010. doi:10.1007/s00208-010-0619-yGoogle Scholar
[7] Kumar Murty, V. and Ram Murty, M., Transcendental values of class group L-functions, II. Proc. Amer. Math. Soc., to appear.Google Scholar
[8] Lang, S., Algebra. Revised third edition, Graduate Texts in Mathematics 211, Springer-Verlag, New York, 2002.Google Scholar
[9] Milnor, J., On polylogarithms, Hurwitz zeta functions, and their Kubert identities. Enseignement Math. (2) 29(1983), 281–322.Google Scholar
[10] Montgomery, H. L. and Vaughan, R. C., Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.Google Scholar
[11] Ram Murty, M. and Saradha, N., Special values of the polygamma functions. Int. J. Number Theory 5(2009), 257–270. doi:10.1142/S1793042109002079Google Scholar
[12] Okada, T., On an extension of a theorem of S. Chowla. Acta Arith. 38(1980/81), 341–345.Google Scholar
[13] Terasoma, T., Mixed Tate motives and multiple zeta values. Invent. Math. 149(2002), 339–369. doi:10.1007/s002220200218Google Scholar
[14] Zagier, D., Values of zeta functions and their applications. In: First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel, 1994, 497–512.Google Scholar