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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Evaluation of the infinite series $ \sum\sp \infty\sb {n=1,(\frac np)=1}(\frac np)\sb 4{}n\sp {-1}$


Authors: Kenneth Hardy and Kenneth S. Williams
Journal: Proc. Amer. Math. Soc. 109 (1990), 597-603
MSC: Primary 11M20; Secondary 11L05, 11N13
MathSciNet review: 1019275
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p$ be a prime $ \equiv 1(\bmod 4)$. The value of the infinite series $ \sum\nolimits_{_{(\tfrac{n}{p}) = 1}^{n = 1}}^\infty {{{(\tfrac{n}{p})}_4}{n^{ - 1}}} $ is given in finite terms.


References [Enhancements On Off] (What's this?)

  • [1] R. L. Long, Algebraic number theory, Marcel Dekker, New York and Basel, 1977. MR 0469888 (57:9668)
  • [2] J. H. Loxton, Some conjectures concerning Gauss sums, J. Reine Angew. Math. 297 (1978), 153-158. MR 0460250 (57:245)
  • [3] C. R. Mathews, Gauss sums and elliptic functions. II. The quartic sum, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France, April 1979 (IHES/M/79/267).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1019275-7
PII: S 0002-9939(1990)1019275-7
Article copyright: © Copyright 1990 American Mathematical Society