Statistical Behaviour and Universality Properties of the Riemann Zeta Function and Other Allied Dirichlet Series.

Date of Submission

December 1981

Date of Award

Winter 12-12-1982

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Master's Dissertation

Degree Name

Master of Technology

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Babu, G. Jogesh (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

0.1 A brief review of the literature : The Riemann Zeta function (z) is defined for c omplex z with Re (z) > 1 by the seriesζ(z)∞∑n=1n-3 and thence by analytic continuation it is defined as a meromorphic function on the entire complex plane with a simple polc at z = 1. In his famous paper (44) of 1859, Bernhard Riemann inauguratec the study of as a f unction of a complex variable (Euler had already consi dered it f or specific real values) by obtaining this analytic continuation. He also exhibited the intimate connection that obtains botween the positi on of the complex zeros of the Zete function and the distribution of the prime numbers. He also establish d the functional eq ation satisfied by this Zeta func- tion. In view of this functi.onal equation (in conjunction with the Euler product formala (z) = TTp(1-p-z)-1 for Re (z) > 1, where the product is over all primes p), Zeta vanishes at the points z = -2,-4,... (these being the so called trivial zeros) and all the other zeros are complex numbers lying in the criti- cal stripp" o≤Re(z)≤1 and they are synmetrically situated about the critical line Re (z) =½. This observation led Riemann to conjecture that all the nontrivial zeros are indeed on the critical line (for a beautiful discussion of what exactly might have motivated this conjecture, see 23, pp.164-166]). This is the celebrated Riemann bypothesis.Due to its paramount significance for the theory of the Ä‘istribution of the primos, the Riemann hypothesis occupies a central position in pure mathematics. In 1896, J. Hadamard and de la Vallee Poussin inde pendently established that the boundary of the critical strip is free from Zeta zeros. At the very onset of further investigation it was oticed that there is a close connection between the distribution of the Zeta zeros and the growth rate of the Zeta function with increasing imaginery part of the argument. Accordingly the main stream of research in analytic number theory has proceeded towards obtaining more and more refined growth estimates (of the Zeta function) and zero- density estimates (of the proportion of Zeta zeros lying ta the right of the critical line). For a representative sample of the results obtained in this direction, see 42 . Though many pro- minent mathematicians continue to contribute to this line of enquiry, and though it has had profound implications for number the ory, it must be admitted that this piecemeal approach has failed to make any qualitatively significant dent in the prablem of the truth or falsity of the Riemann hypothesis itself. Today things remain more or less where Riemann, Hadamard and de la Vallee Poussin left them. For instance, it is not yet known if any proper substrip of the critical strip can be free of Zeta zerosA second line of enquiry into the Rienann Zeta function was opened up in a series of brilliant papers by Harald Bohr. The essence of this approach is to take a global view of the Zeta function and study its general value distribution in place of prematurely restricting one self to a consideration of it zeros alone.


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