Koshliakov zeta functions I: Modular relations

We examine an unstudied manuscript of N.S. Koshliakov over 150 pages long and containing the theory two interesting generalizations ζp(s) ηp(s) Riemann zeta function ζ(s), which we call functions. His has its genesis in a problem analytical heat distribution was analyzed by him. In this paper, further build upon his obtain new modular relations setting functions, each gives infinite family identities, one for p∈R+. The first is generalization Ramanujan's famous formula ζ(2m+1) second elegant extension relation on page 220 Lost Notebook. Several corollaries applications these are obtained including representation ζ(4m+3).