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25
On Kummer's Conjecture
, 2001
"... Kummer conjectured the asymptotic behavior of the first factor of the class number of a cyclotomic field. If we only ask for upper and lower bounds of the order of growth predicted by Kummer, then this modified Kummer conjecture is true for almost all primes. ..."
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Kummer conjectured the asymptotic behavior of the first factor of the class number of a cyclotomic field. If we only ask for upper and lower bounds of the order of growth predicted by Kummer, then this modified Kummer conjecture is true for almost all primes.
Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
DIVISORS OF SHIFTED PRIMES
"... Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the correspo ..."
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Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants. 1.
Artin’s conjecture; Unconditional approach and elliptic analogue
 Master’s thesis
, 2008
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, I have explored the different approach ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, I have explored the different approaches towards proving Artin’s ‘primitive root ’ conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by E. Artin in the year 1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on the assumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid of the assumption and it seemed quite a daunting task. In 1983, the pioneering attempt was made by R. Gupta and M. Ram Murty, who proved unconditionally that there exists a specific set of 13 distinct numbers such that for at least one of them, the conjecture is true. Along the same line, using sieve theory, D. R. HeathBrown reduced this set down to 3 distinct primes in the year 1986. This is
Prime numbers and Lfunctions
"... Abstract. The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this t ..."
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Abstract. The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk I will show a panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory.
Different Approaches to the Distribution of Primes
 MILAN JOURNAL OF MATHEMATICS
, 2009
"... In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction. ..."
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In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction.
GENERALIZED AND RESTRICTED MULTIPLICATION TABLES OF INTEGERS
, 2010
"... In 1955 Erdős posed the multiplication table problem: Given a large integer N, how many distinct products of the form ab with a ≤ N and b ≤ N are there? The order of magnitude of the above quantity was determined by Ford. The purpose of this thesis is to study generalizations of Erdős’s question in ..."
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In 1955 Erdős posed the multiplication table problem: Given a large integer N, how many distinct products of the form ab with a ≤ N and b ≤ N are there? The order of magnitude of the above quantity was determined by Ford. The purpose of this thesis is to study generalizations of Erdős’s question in two different directions. The first one concerns the kdimensional version of the multiplication table problem: for a fixed integer k ≥ 3 and a large parameter N, we establish the order of magnitude of the number of distinct products n1 · · · nk with ni ≤ N for all i ∈ {1,..., k}. The second question we shall discuss is the restricted multiplication table problem. More precisely, for B ⊂ N we seek estimates on the number of distinct products ab ∈ B with a ≤ N and b ≤ N. This problem is intimately connected with the available information on the distribution of B in arithmetic progressions. We focus on the special and important case when B = Ps = {p + s: p prime} for some fixed s ∈ Z \ {0}. Ford established upper bounds of the expected order of magnitude for {ab ∈ Ps: a ≤ N, b ≤ N}. We prove the corresponding lower bounds thus determining the size of the quantity in question up to multiplicative constants.
In this paper we consider the exponential sum
"... This is the author’s version of the paper. ..."
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