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On Pellarin's $ L$-series


Author: Rudolph Bronson Perkins
Journal: Proc. Amer. Math. Soc. 142 (2014), 3355-3368
MSC (2010): Primary 11M38
DOI: https://doi.org/10.1090/S0002-9939-2014-12080-6
Published electronically: June 16, 2014
MathSciNet review: 3238413
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Abstract: Necessary and sufficient conditions are given for a negative integer to be a trivial zero of a new type of $ L$-series recently discovered by F. Pellarin, and it is shown that any such trivial zero is simple. We determine the exact degree of the special polynomials associated to Pellarin's $ L$-series. The theory of Carlitz polynomial approximations is developed further for both additive and $ \mathbb{F}_q$-linear functions. Using Carlitz's theory we give a generating series for the power sums occurring as the coefficients of the special polynomials associated to Pellarin's series, and a connection is made between the Wagner representation for $ \chi _t$ and the value of Pellarin's $ L$-series at 1.


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  • [1] Greg W. Anderson and Dinesh S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. of Math. (2) 132 (1990), no. 1, 159-191. MR 1059938 (91h:11046), https://doi.org/10.2307/1971503
  • [2] Gebhard Böckle, Global $ L$-functions over function fields, Math. Ann. 323 (2002), no. 4, 737-795. MR 1924278 (2003e:11052), https://doi.org/10.1007/s002080200325
  • [3] G. Böckle,
    The distribution of the zeros of the Goss zeta-function for $ A = \mathbb{F}_2[x, y]/(y^2 + y + x^3 + x + 1)$, Math. Z. 275 (2013), no. 3-4, 835-861. MR 3127039
  • [4] Leonard Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), no. 2, 137-168. MR 1545872, https://doi.org/10.1215/S0012-7094-35-00114-4
  • [5] L. Carlitz, A set of polynomials, Duke Math. J. 6 (1940), 486-504. MR 0001946 (1,324a)
  • [6] David Goss, Fourier series, measures and divided power series in the theory of function fields, $ K$-Theory 2 (1989), no. 4, 533-555. MR 990575 (90i:11138), https://doi.org/10.1007/BF00533281
  • [7] David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR 1423131 (97i:11062)
  • [8] David Goss, Applications of non-Archimedean integration to the $ L$-series of $ \tau $-sheaves, J. Number Theory 110 (2005), no. 1, 83-113. MR 2114675 (2006e:11134), https://doi.org/10.1016/j.jnt.2004.05.014
  • [9] D. Goss,
    On the $ {L}$-series of F. Pellarin,
    J. Number Theory 133 (2013), no. 3, 955-962.
    doi:10.1016/j.jnt.2011.12.001. MR 2997778
  • [10] F. Pellarin,
    Values of certain $ {L}$-series in positive characteristic,
    Ann. of Math. (2) 176 (2012), 2055-2093. MR 2979866
  • [11] Jeffrey T. Sheats, The Riemann hypothesis for the Goss zeta function for $ \mathbf {F}_q[T]$, J. Number Theory 71 (1998), no. 1, 121-157. MR 1630979 (99b:11127), https://doi.org/10.1006/jnth.1998.2232
  • [12] Carl G. Wagner, Interpolation series for continuous functions on $ \pi $-adic completions of $ {\rm GF}(q,\,x)$, Acta Arith. 17 (1970/1971), 389-406. MR 0282973 (44 #207)

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Additional Information

Rudolph Bronson Perkins
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: perkins@math.osu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12080-6
Received by editor(s): December 29, 2011
Received by editor(s) in revised form: February 21, 2012, and October 18, 2012
Published electronically: June 16, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society

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