This is a survey article covering certain important mathematical contributions of K. Ramachandra to the theory of the Riemann zeta-function and their impact on current research.

Hardy's theorem for the Riemann zeta-function ζ(s) says that it admits infinitely many complex zeros on the line (s) = 1 2. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.

An arithmetic function f is a sieve function of range Q, if its Eratosthenes transform g = f * µ is supported in [1, Q]∩N, where g(q) ε q ε , ∀ε > 0. Here, we study the distribution of f over the so-called short arithmetic bands 1≤a≤H {n ∈ (N, 2N ] : n ≡ a (mod q)}, with H = o(N), and give applications to both the correlations and to the so-called weighted Selberg integrals of f , on which we have concentrated our recent research.

We study Ramanujan-Fourier series of certain arithmetic functions of two variables. We generalize Delange's theorem to the case of arithmetic functions of two variables and give sufficient conditions for pointwise convergence of Ramanujan-Fourier series of arithmetic functions of two variables. We also give several examples which are not obtained by trivial generalizations of results on Ramanujan-Fourier series of functions of one variable.

Lectures on the Riemann Zeta Function, by Henryk Iwaniec, University Lecture Series, Volume: 62, American Mathematical Society, Providence, RI, 2014, viii+119 pp., Softcover, ISBN 978-1-4704-1851.