Character analogues of cohen-type identities and related voronoi summation formulas

Abstract

Summation formulas play a vital role in analytic number theory. Several kinds of summation formulas exist, like the Poisson summation formula, Abel summation formula, Euler-Maclaurin formula, etc. In 1904, G. F. Voronoi proved that the error term in the Dirichlet divisor problem can be expressed in terms of infinite series involving Bessel functions. Additionally, he offered a broader version of the above summation formulas involving a test function f , where f (t) is a function of bounded variation. Consequently, he deduced a better bound for the error term in the Dirichlet divisor problem at that time. Following Voronoi’s astounding discovery, other number theorists like A. L. Dixon,W. L. Ferrar, J. R. Wilton, Koshliakov, M. Jutila etc looked into the formula and offered proofs under different conditions on the function f (x). Apart from its con-nection to different fields of mathematics, Voronoi-type summation formulas also have some applications in physics, especially in quantum graph theory. In 2014, B. C. Berndt and A. Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying Ramanujan’s type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, S. Kim extended the definition of twisted divisor sums to twisted sums of divisor functions . Here, we study identities associated with the aforementioned weighted divisor functions and the modified K-Bessel function in light of recent results obtained by D. Banerjee and B. Maji. Moreover, we provide a new expression for L(1, χ)from which the positivity of L(1, χ) for any real primitive character χ is established, which is important is the proof of Prime number theorems in arithmetic progression. In addition, we deduce Cohen-type identities and then exhibit the Voronoi-type summation formulas for them. Additionally, we discuss an equivalent version of the aforementioned results in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, we provide an identity for r6(n), which is analo-gous to Hardy’s famous result where r6(n) denotes the number of representations of natural number n as a sum of six squares.