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ABC Implies No "Siegel Zeros" For LFunctions Of Characters With Negative Discriminant
 Inventiones Math
"... this paper we will apply the uniform abcconjecture to the very large solutions of Diophantine equations that arise from modular functions and deduce a lower bound for the class number of imaginary quadratic fields. This extends an idea of Chowla [1,2] who indicated, via a conjecture of Hall, how un ..."
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Cited by 6 (1 self)
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this paper we will apply the uniform abcconjecture to the very large solutions of Diophantine equations that arise from modular functions and deduce a lower bound for the class number of imaginary quadratic fields. This extends an idea of Chowla [1,2] who indicated, via a conjecture of Hall, how unlikely it is that
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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Cited by 4 (3 self)
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
Rabinowitsch Revisited
"... this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is wellknown (see [5]) that if the class number of some imaginary quadratic field with large discriminant ..."
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this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is wellknown (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet Lfunction which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n
SMALL GAPS BETWEEN PRIMES II (PRELIMINARY)
"... Abstract. We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (pn+1 − pn) (1.1) lim inf n→ ∞ log pn(log log pn) −1 < ∞. log log log log pn Further we show that supposing the ..."
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Abstract. We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (pn+1 − pn) (1.1) lim inf n→ ∞ log pn(log log pn) −1 < ∞. log log log log pn Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ Xϑ with any level ϑ>1/2 we have bounded differences between consecutive primes infinitely often: (1.2) lim inf n→ ∞ (pn+1 − pn) ≤ C(ϑ) with a constant C(ϑ) depending only on ϑ. If the Bombieri–Vinogradov theorem holds with a level ϑ>20/21, in particular if the Elliott–Halberstam conjecture holds, then we obtain (1.3) lim inf n→ ∞ (pn+1 − pn) ≤ 20, that is pn+1 − pn ≤ 20 for infinitely many n. Inequalities (1.2)–(1.3) will follow from the even stronger following result Theorem A. Suppose the Bombieri–Vinogradov theorem is true for Q ≤ Xϑ with some ϑ>1/2. Then there exists a constant C ′ (ϑ) such that any admissible ktuple contains at least two primes for any (1.4) k ≥ C ′ (ϑ) if ϑ>1/2, where C ′ (ϑ) is an explicitly calculable constant depending only on ϑ. Further we have at least two primes for (1.5) k =7 if ϑ>20/21. Remark. For the definition of admissibility see (2.2) below. We will show some more general results for the quantity (ν is a given positive integer) (1.6) Eν = lim inf n→∞ pn+ν − pn log pn
Uniform bounds for the least almostprime primitive root, submitted
"... A recurring theme in number theory is that multiplicative and additive properties of integers are more or less independent of each other, the classical result in this vein being Dirichlet’s theorem on primes in arithmetic progressions. Since the set of primitive roots to a given modulus is a union o ..."
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A recurring theme in number theory is that multiplicative and additive properties of integers are more or less independent of each other, the classical result in this vein being Dirichlet’s theorem on primes in arithmetic progressions. Since the set of primitive roots to a given modulus is a union of arithmetic progressions, it is natural to study the distribution of
PRIMES IN TUPLES II
, 710
"... Abstract. We prove that liminf n→∞ pn+1 − pn √ log pn(log log pn) 2 where pn denotes the n th prime. Since on average pn+1 −pn is asymptotically log pn, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the se ..."
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Abstract. We prove that liminf n→∞ pn+1 − pn √ log pn(log log pn) 2 where pn denotes the n th prime. Since on average pn+1 −pn is asymptotically log pn, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p − p ′ between primes which includes the small gap result above. 1.
DUKE MATHEMATICAL JOURNAL Vol. 111, No. 3, c ○ 2002 SOME REMARKS ON LANDAUSIEGEL ZEROS
"... In this paper we show that, under the assumption that all the zeros of the Lfunctions under consideration are either real or lie on the critical line, one may considerably improve on the known results on LandauSiegel zeros. 1. ..."
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In this paper we show that, under the assumption that all the zeros of the Lfunctions under consideration are either real or lie on the critical line, one may considerably improve on the known results on LandauSiegel zeros. 1.
Typeset by A ST X RABINOWITSCH REVISITED
"... In the late eighteenth century both Euler and Legendre noticed that n2 + n +41 is prime for n =0, 1, 2...39,and remarked that there are few polynomials with such small degree and coefficients that give such a long string of consecutive prime ..."
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In the late eighteenth century both Euler and Legendre noticed that n2 + n +41 is prime for n =0, 1, 2...39,and remarked that there are few polynomials with such small degree and coefficients that give such a long string of consecutive prime