Rook theory, compositions, and zeta functions

Author:
James Haglund

Journal:
Trans. Amer. Math. Soc. **348** (1996), 3799-3825

MSC (1991):
Primary 11M41, 05A15

DOI:
https://doi.org/10.1090/S0002-9947-96-01662-5

MathSciNet review:
1357880

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in . Some identities in the ring of formal power series involving rook theory and continued fractions are developed.

**[All]**K. Alladi,*Some New Observations on the Göllnitz-Gordon and Rogers-Ramanujan Identities*, Trans. Amer. Math. Soc.**347**(1995), 897--914. MR**95h:11109****[An1]**G. E. Andrews,*The Theory of Partitions*, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, Reading, Mass., 1976. MR**58:27738****[An2]**G. E. Andrews,*The Theory of Compositions (I): The Ordered Factorizations of n and a conjecture of C. Long*, Canad. Math. Bull.**18**(1975), 479-484. MR**54:7363****[An3]**G. E. Andrews,*The Theory of Compositions (II): Simon Newcomb's Problem*, Utilitas Math.**7**(1974), 33-54. MR**54:7364a****[An4]**G. E. Andrews,*The Theory of Compositions (III): the MacMahon Formula and the Stanton-Cowan Numbers*, Utilitas Math.**9**(1976), 283-290. MR**54:7364b****[Car]**L. Carlitz,*Extended Bernoulli and Eulerian numbers*, Duke Math. J.**31**(1964), 667-689. MR**29:5796****[Cra]**H. Cramér,*On the order of magnitude of the difference between consecutive prime numbers*, Acta Arith.**2**(1936), 23-46.**[D-R]**J. F. Dillon and D. P. Roselle,*Simon Newcomb's Problem*, SIAM J. Appl. Math.**17**(1969), 1086-1093. MR**41:1553****[F-S]**D. C. Foata and M. P. Schützenberger,*On the rook polynomials of Ferrers relations*, Colloq. Math. Soc. Janos Bolyai, 4, Combinatorial Theory and its Applications, vol. 2, North-Holland, Amsterdam, 1970, pp. (413--436). MR**50:12738****[Gra]**A. Granville,*Harald Cramér and the distribution of prime numbers*, Scand. Actuar. J.**1995**, 12--28. CMP**95:17****[GJW]**J. Goldman, J. Joichi, and D. White,*Rook theory I: Rook equivalence of Ferrers boards*, Proc. Amer. Math. Soc.**52**(1975), 485-492. MR**55:2590****[H-W]**G. H. Hardy and E. M. Wright,*An Introduction to the Theory of Numbers*, Fifth Edition, Clarendon Press, Oxford, 1979. MR**81i:10002****[Ha1]**J. Haglund,*Rook Placements, Compositions, and Permutations of Vectors*, Doctoral Dissertation, University of Georgia, Athens, Georgia, 1993.**[Ha2]**J. Haglund,*Compositions and rook placements*, preprint, 1994.**[Ha3]**J. Haglund,*Compositions and -rook polynomials*, in preparation.**[K-R]**I. Kaplansky and J. Riordan,*The problem of the rooks and its applications*, Duke Math. J.**13**(1946), 259-268. MR**7:508d****[Mac]**P. A. MacMahon,*Combinatory Analysis*, Vol. 1, Cambridge University Press, 1915.**[Per]**O. Perron,*Die Lehre von den Kettenbrüchen*, Chelsea, New York, 1950. MR**12:254b****[Raw]**D. Rawlings,*The Simon Newcomb Problem*, Linear and Multilinear Algebra**10**(1981), 253-260. MR**83k:05009****[Rio]**J. Riordan,*An Introduction to Combinatorial Analysis*, John Wiley, New York, 1958. MR**20:3077****[RNT]**Richard K. Guy (editor),*Reviews in Number Theory 1973-83*, Vol. 1a, Amer. Math. Soc., Providence, RI, 1984.**[Sat]**J. Satoh,*-Analogue of Riemann's -function and -Euler numbers*, J. Number Theory**31**(1989), 346-362. MR**90d:11132****[Si1]**R. Simion,*On Compositions of Multisets*, Doctoral Dissertation, The University of Pennsylvania, 1981.**[Si2]**R. Simion,*A multi-indexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences*, J. Combin. Theory Ser. A**36**(1984), 15-22. MR**85e:05015****[Sta]**R. P. Stanley,*Enumerative Combinatorics: Volume I*, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986. MR**87j:05003****[Tic]**E. C. Titchmarsh,*The Theory of the Riemann Zeta-Function*, Second Edition, Clarendon Press, 1986. MR**88c:11049**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
11M41,
05A15

Retrieve articles in all journals with MSC (1991): 11M41, 05A15

Additional Information

**James Haglund**

Affiliation:
Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana, IL 61801

Email:
jhaglund@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01662-5

Keywords:
Riemann zeta function,
Riemann hypothesis,
continued fraction,
composition,
rook theory,
Euler product

Received by editor(s):
January 20, 1995

Received by editor(s) in revised form:
November 6, 1995

Article copyright:
© Copyright 1996
American Mathematical Society