Rook theory, compositions, and zeta functions

Author:
James Haglund

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

MSC (1991):
Primary 11M41, 05A15

MathSciNet review:
1357880

Full-text PDF Free Access

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]**Krishnaswami Alladi,*Some new observations on the Göllnitz-Gordon and Rogers-Ramanujan identities*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 897–914. MR**1284910**, 10.1090/S0002-9947-1995-1284910-4**[An1]**George E. Andrews,*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013****[An2]**George E. Andrews,*The theory of compositions. I. The ordered factorizations of 𝑛 and a conjecture of C. Long*, Canad. Math. Bull.**18**(1975), no. 4, 479–484. MR**0419341****[An3]**George E. Andrews,*The theory of compositions. II. Simon Newcomb’s problem*, Utilitas Math.**7**(1975), 33–54. MR**0419342****[An4]**George E. Andrews,*The theory of compositions. III. The MacMahon formula and the Stanton-Cowan numbers*, Utilitas Math.**9**(1976), 283–290. MR**0419343****[Car]**L. Carlitz,*Extended Bernoulli and Eulerian numbers*, Duke Math. J.**31**(1964), 667–689. MR**0168534****[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**0256898****[F-S]**D. Foata and M. P. Schützenberger,*On the rook polynomials of Ferrers relations*, Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969) North-Holland, Amsterdam, 1970, pp. 413–436. MR**0360288****[Gra]**A. Granville,*Harald Cramér and the distribution of prime numbers*, Scand. Actuar. J.**1995**, 12--28. CMP**95:17****[GJW]**Jay R. Goldman, J. T. Joichi, and Dennis E. White,*Rook theory. I. Rook equivalence of Ferrers boards*, Proc. Amer. Math. Soc.**52**(1975), 485–492. MR**0429578**, 10.1090/S0002-9939-1975-0429578-4**[H-W]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR**568909****[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]**Irving Kaplansky and John Riordan,*The problem of the rooks and its applications*, Duke Math. J.**13**(1946), 259–268. MR**0016082****[Mac]**P. A. MacMahon,*Combinatory Analysis*, Vol. 1, Cambridge University Press, 1915.**[Per]**Oskar Perron,*Die Lehre von den Kettenbrüchen*, Chelsea Publishing Co., New York, N. Y., 1950 (German). 2d ed. MR**0037384****[Raw]**Don Rawlings,*The (𝑞,𝑟)-Simon Newcomb problem*, Linear and Multilinear Algebra**10**(1981), no. 3, 253–260. MR**630153**, 10.1080/03081088108817417**[Rio]**John Riordan,*An introduction to combinatorial analysis*, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR**0096594****[RNT]**Richard K. Guy (editor),*Reviews in Number Theory 1973-83*, Vol. 1a, Amer. Math. Soc., Providence, RI, 1984.**[Sat]**Junya Satoh,*𝑞-analogue of Riemann’s 𝜁-function and 𝑞-Euler numbers*, J. Number Theory**31**(1989), no. 3, 346–362. MR**993908**, 10.1016/0022-314X(89)90078-4**[Si1]**R. Simion,*On Compositions of Multisets*, Doctoral Dissertation, The University of Pennsylvania, 1981.**[Si2]**Rodica Simion,*A multi-indexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences*, J. Combin. Theory Ser. A**36**(1984), no. 1, 15–22. MR**728500**, 10.1016/0097-3165(84)90075-X**[Sta]**Richard P. Stanley,*Enumerative combinatorics. Vol. I*, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986. With a foreword by Gian-Carlo Rota. MR**847717****[Tic]**E. C. Titchmarsh,*The theory of the Riemann zeta-function*, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR**882550**

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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