Number Theory

1. Front Matter

   Pages i-vii

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2. A Centennial History of the Prime Number Theorem

   • Tom M. Apostol

   Pages 1-14

3. Non-homogeneous Problems: Conjectures of Minkowski and Watson

   • R. P. Bambah, V. C. Dumir, R. J. Hans-Gill

   Pages 15-41

4. On the Oscillation Theorems of Pringsheim and Landau

   • Paul T. Bateman, Harold G. Diamond

   Pages 43-54

5. Modular Equations in Ramanujan’s Lost Notebook

   • Bruce C. Berndt

   Pages 55-74

6. The abc-conjecture

   • Jerzy Browkin

   Pages 75-105

7. On Values of Linear and Quadratic Forms at Integral Points

   • S. G. Dani

   Pages 107-119

8. Variants of the Second Borel-Cantelli Lemma and their Applications in Metric Number Theory

   • Glyn Harman

   Pages 121-140

9. Pythagorean Triples

   • Edmund Hlawka

   Pages 141-155

10. Integer Points in Plane Regions and Exponential Sums

    • M. N. Huxley

    Pages 157-166

11. Artin’s Conjecture for Polynomials Over Finite Fields

    • Erik Jensen, M. Ram Murty

    Pages 167-181

12. Continuous Homomorphisms as Arithmetical Functions, and Sets of Uniqueness

    • I. Kátai

    Pages 183-200

13. Hamburger’s Theorem on ζ(s) and the Abundance Principle for Dirichlet Series with Functional Equations

    • Marvin I. Knopp

    Pages 201-216

14. A Survey of Number Theory and Cryptography

    • Neal Koblitz

    Pages 217-239

15. Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions

    • Kohji Matsumoto

    Pages 241-286

16. Algebraic Curves Over Finite Fields with many Rational Points and their Applications

    • Harald Niederreiter, Chaoping Xing

    Pages 287-300

17. A Report on Artin’s Holomorphy Conjecture

    • Dipendra Prasad, C. S. Yogananda

    Pages 301-314

18. Siegel’s Main Theorem for Quadratic Forms

    • S. Raghavan

    Pages 315-324

19. Pfister’s Work on Sums of Squares

    • A. R. Rajwade

    Pages 325-349

20. Notes on the Prime Number Theorem-I

    • K. Ramachandra

    Pages 351-370

The Indian National Science Academy on the occasion ofthe Golden Jubilee Celebration (Fifty years of India's Independence) decided to publish a number of monographs on the selected fields. The editorial board of INS A invited us to prepare a special monograph in Number Theory. In reponse to this assignment, we invited several eminent Number Theorists to contribute expository/research articles for this monograph on Number Theory. Al­ though some ofthose invited, due to other preoccupations-could not respond positively to our invitation, we did receive fairly encouraging response from many eminent and creative number theorists throughout the world. These articles are presented herewith in a logical order. We are grateful to all those mathematicians who have sent us their articles. We hope that this monograph will have a significant impact on further development in this subject. R. P. Bambah v. C. Dumir R. J. Hans-Gill A Centennial History of the Prime Number Theorem Tom M. Apostol The Prime Number Theorem Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: x (I) K(X) '" -I - as x --+ 00, ogx and Pn '" n log n as n --+ 00. (2) In (1), K(X) denotes the number of primes P ::s x for any x > O.

• algebra
• arithmetic
• boundary element method
• cryptography
• diophantine equation
• finite field
• form
• graphs
• homomorphism
• number theory
• polynomial
• prime number
• quadratic form
• story
• theorem

• Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India

  R. P. Bambah, V. C. Dumir, R. J. Hans-Gill

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