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Ramachandra, K.
- On a Discrete Mean Value Theorem for ζj
Authors
1 The Institute for Advanced Study, Princeton, New Jersey, US
Source
The Journal of the Indian Mathematical Society, Vol 36, No 3-4 (1972), Pagination: 307-316Abstract
In the course of the proof of the theorem, I also give a proof of my result mentioned above. We first state two corollaries to the theorem above.- Two Remarks on a Result of Ramachandra
Authors
1 Tata Institute of Fundamental Research, Bombay 400 005, IN
Source
The Journal of the Indian Mathematical Society, Vol 38, No 1-4 (1974), Pagination: 395-397Abstract
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.
- Simplest, Quickest and Self-Contained Proof that1/2≤θ≤1(θ Being the Least Upper Bound of the Real Parts of the Zeros of ζ(s))
Authors
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay-400005, IN
Source
The Journal of the Indian Mathematical Society, Vol 61, No 1-2 (1995), Pagination: 7-12Abstract
We will prove that ζ(s) (s=σ+it), defined byζ(s)=Σn-s (σ>1), (1)
=(n-s-(n+u)-s)du+1/s-1 (σ>0),(2)
=(1-p-s)-1(σ>1),(1)
has infinity of complex zeros in a σ≥1/2-δ for every constant δ>0. (In (3) the product runs over all primes p). Let θ denote the least upper bound of the real parts of the zeros of ζ(s). Then by (3) we have trivially θ≤1 and by what we will prove it follows that θ≥1/2. These results are not new. But the merit of our proof is the fact that apart from using Cauchy’s Theorem for certain rectangles, we use only the simple facts given by (1), (2) and (3). The proof is simpler than the one given in [1]. Without complicating the proof we prove the following theorem.
- Notes on Prime Number Theorem-II
Authors
1 Nat. Hist, of Adv. Studies. 1.1. Sc. Campus, Bangalore-560012, IN
2 TIFR, Homi Bhabha Road, Colaba, Munibai-400 005, IN
3 Matscicnce, Tharamani P.O-600 113, Chennai, Tamil Nadu, IN
Source
The Journal of the Indian Mathematical Society, Vol 72, No 1-4 (2005), Pagination: 13-18Abstract
In a series of papers, the Soviet mathematician I.M. Vinogradov developed a very important method of dealing with estimation of trigonometric sums. (See Chapter VI of [ECT] and [ECT, DRHB]).- Some Remarks on the Mean-Value of the Riemann Zeta-Function and other Dirichlet Series-IV
Authors
1 School of Mathematics, Tata Institute of Fundamental Research, Bombay-400005, IN
Source
The Journal of the Indian Mathematical Society, Vol 60, No 1-4 (1994), Pagination: 107-122Abstract
The object of this paper is to prove the following theorem.
Theorem 1. Let α and k be real numbers subject to 1/2+q(log H)-1≤α≤2 (where q is a positive integer to be defined presently) and δ≤k≤δ-1 where δ is any positive constant.
- Notes on the Riemann Zeta-Function
Authors
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, IN
Source
The Journal of the Indian Mathematical Society, Vol 57, No 1-4 (1991), Pagination: 67-77Abstract
In a recent paper [2] R. Balasubramanian and K. Ramachandra proved results likemax |ζ(1/2+it)|>t0-δ
where ∈ is an arbitrary positive constant, t0 exceeds a positive constant depending on ∈ and C(∈) depends on ∈. In fact their results were very general and they could replace ζ(1/2+it) by F(σ+it) for very general Dirichlet series P(s), and prove (1) for F(σ+it). In this paper we record three theorems and indicate their proof. These are probably well-known to the experts in this field or at least within their easy reach. But the results are so interesting that they deserve to be printed.
- An Easy Transcendence Measure for e
Authors
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, IN
Source
The Journal of the Indian Mathematical Society, Vol 51, No 1-2 (1987), Pagination: 111-116Abstract
The object of this note is to give a simple proof of the following theorem.Theorem 1. Let n≥1 and a0,a1, ..., an be integers of which an≠0 and max|a1|≤H where H exceeds a constant Ho(n) depending only on n. Then
|a0+a1e+...+anen|≥H-cn
where C is an absolute constant.