Character analogues of theorems of Ramanujan, Koshliakov and Guinand

We derive analogues of theorems of Ramanujan, Koshliakov, and Guinand involving primitive characters and modified Bessel functions. As particular examples, transformation formulas involving the Legendre symbol and sums-of-divisors functions are established.

[1]  K. Williams,et al.  Gauss and Jacobi sums , 2021, Mathematical Surveys and Monographs.

[2]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[3]  Bruce C. Berndt,et al.  Hecke's Theory of Modular Forms and Dirichlet Series , 2007, Monographs in Number Theory.

[4]  T. Apostol Introduction to analytic number theory , 1976 .

[5]  W. J. Harvey,et al.  THE COMPLEX ANALYTIC THEORY OF TEICHMÜLLER SPACES (Canadian Mathematical Society Series of Monographs and Advanced Texts) , 1991 .

[6]  B. Berndt,et al.  Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook , 2008 .

[7]  Bruce C. Berndt,et al.  Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory , 1975 .

[8]  George E. Andrews,et al.  The lost notebook and other unpublished papers , 1988 .

[9]  G. N. Watson SOME SELF-RECIPROCAL FUNCTIONS , 1931 .

[10]  Gian-Carlo Rota,et al.  Theory and application of special functions , 1977 .

[11]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[12]  Bruce C. Berndt Character analogues of the poisson and Euler-MacLaurin summation formulas with applications , 1975 .

[13]  Mourad E. H. Ismail,et al.  Theory and Applications of Special Functions , 2005 .

[14]  V. Rane Analogues of Euler and Poisson summation formulae , 2003, math/0310285.

[15]  Bruce C. Berndt Periodic Bernoulli Numbers, Summation Formulas and Applications , 1975 .