Two Remarks on a Result of Ramachandra

Jump To References Section

Authors

  • Tata Institute of Fundamental Research, Bombay 400 005 ,IN
  • Tata Institute of Fundamental Research, Bombay 400 005 ,IN

Abstract

Improving on the results of Montgomery [3] and Huxley [1], Ramachandra proved (see Lemma 4 of [5]) the following large value theorem:

THEOREM 1. Let an = an(N) (n = N+1, . . . , 2N) be complex numbers subject to the condition max |an| = O(Nε) for every ε > 0. Suppose that n N does not exceed a fixed power of T to be defined. Let V be a positive number such that V+1/v= O(Tε)for every ε > 0. Let Sr (r = 1, 2, ...,R; R≥2) be a set of distinct complex numbers Sr = σr + itr and let min σr = σ, 3/4 ≤ σ ≤ 1,

max tr - min tr + 20 = T, min |tr - tr|≥(log T)2.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Published

1974-12-01

How to Cite

Balasubramanian, R., & Ramachandra, K. (1974). Two Remarks on a Result of Ramachandra. The Journal of the Indian Mathematical Society, 38(1-4), 395–397. Retrieved from https://informaticsjournals.com/index.php/jims/article/view/16716

 

References

M. N. HUXLEY : On the Difference between consecutive primes. Invent, Math., 15(1972), 164-170.

M. JUTILA: On a density theorem of H.L. Montgomery for L-functions, Annates Acad. Sci., Fennicae., Series A, I. Mathematica 520 (1972), 1-13.

H.L.MONTGOMERY: Mean and large values of Dirichlet Polynomials, Invent. Math 8(1969), 334-345.

H. L. MONTGOMERY: Topics in Multiplicative Number Theory. Lecturenotes on Mathematics Springer-Verlag (1971).

K. RAMACHANDRA: Some new density estimates for the zeros of the Riemann zeta-function, (to appear).