The Journal of the Indian Mathematical Society

K. Ramachandra ¹

Affiliations
1 The Institute for Advanced Study, Princeton, New Jersey, US

Abstract

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.

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R. Balasubramanian ¹, K. Ramachandra ¹

Affiliations
1 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 a_n = a_n(N) (n = N+1, . . . , 2N) be complex numbers subject to the condition max |a_n| = O(N^ε) 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(T^ε)for every ε > 0. Let S_r (r = 1, 2, ...,R; R≥2) be a set of distinct complex numbers S_r = σ_r + it_r and let min σ_r = σ, 3/4 ≤ σ ≤ 1,

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

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K. Ramachandra ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay-400005, IN

Abstract

We will prove that ζ(s) (s=σ+it), defined by
ζ(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.

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K. Ramachandra ¹, A. Sankaranarayanan ², K. Srinivas ³

Affiliations
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

Abstract

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]).

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K. Ramachandra ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Bombay-400005, IN

Abstract

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.

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K. Ramachandra ¹, A. Sankaranarayanan ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, IN

Abstract

In a recent paper [2] R. Balasubramanian and K. Ramachandra proved results like
max |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.

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K. Ramachandra ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, IN

Abstract

The object of this note is to give a simple proof of the following theorem.
Theorem 1. Let n≥1 and a₀,a₁, ..., a_n be integers of which a_n≠0 and max|a₁|≤H where H exceeds a constant H_o(n) depending only on n. Then
|a₀+a₁e+...+a_neⁿ|≥H^-cn
where C is an absolute constant.

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