## Abstract

Although no longer affiliated with the TIFR, DDK persisted in his prime obsession, repeating and refining the basic arguments set forth in his earlier papers. The Hungarian mathematician A. Rényi reviewed this paper in Mathematical Reviews and noted that “Neither in this paper nor in his previous paper did the author succeed in proving his hypothesis, nor in deducing from it the Riemann hypothesis.” The “Kosambi hypothesis” (see the discussion on pages 7–9) is, according to Rényi “even more difficult than the problem of the validity of the Riemann hypothesis. As a matter of fact, no obvious method exists to prove the author’s hypothesis even under the assumption of the Riemann hypothesis.” Theorem 1 of this paper, therefore, is not proven.

## Keywords

Iterate Logarithm Riemann Hypothesis Basic Argument Euler Product Prime Number Theorem## References

- 1.E.C. Titchmarsh,
*The Theory of Functions*(Oxford, 1932), p. 152.Google Scholar - 2.E.C. Titchmarsh
*The Zeta-Function of Riemann*(Oxford, 1951), pp. 214–22. Chapters xiii-xiv give the consequences of \(RH\).Google Scholar - 3.K. Prachar
*Primzahlverteilung*, (Berlin, 1957), pp. 247–55.Google Scholar - 4.Littlewood, J.E. Sur la distribution des nombres premiers.
*Comptes Rendus*, (Paris, 1914), vol 158, (pp. 72–1869). and note 3 above.Google Scholar - 5.K. Prachar,
*Loc. cit.,*Chapter ii.Google Scholar - 6.
- 7.G. Ricci, Rendiconti, Atti. Acad. Naz. Lincei,
**8**, 96–192 and 51–347 (1954–55). Prof. P. Erdős was kind enough to inform me that the Ricci statement must be emended to: ‘the set of cluster points for the \(y\)-distance between consecutive primes is of positive measure’. Even this suffices to prove Lemma 2, except that in this case, not every displacement however small need lead to a different sequence throughout the \(u\)-interval. This will be replaced by a set of sub-intervals of the \(u\)-interval, whose total length remains positive, and may then be mapped upon (0, 1).Google Scholar - 8.G.H. Hardy, E.M. Wright,
*An Introduction to the Theory of Numbers*, 2nd edn. (Oxford, 1945), pp. 349–54. Theorem 430.Google Scholar - 9.G.H. Hardy, Wright, E.M.
*Ibid*., Sections 9.12 and 9.13.Google Scholar - 10.W. Feller,
*Introduction to Probability Theory and Its Applications*, vol. 1, (New York, 1950), pp. 161–63. proof for finite base only.Google Scholar - 11.
- 12.W. Feller,
*Ibid.*, p. 154.Google Scholar - 13.D.D. Kosambi, The sampling distribution of primes. Proc. Nat. Acad. Sci. (U.S.A.),
**49**, 20–23 (1963).Google Scholar - 14.A.E. Ingham, Quart. J. Math. (Oxford)
**8**, 255–66 (1937).Google Scholar