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Numerical computations concerning the ERH
 Math. Comp
, 1993
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On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters.
, 1993
"... : We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a s ..."
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Cited by 5 (1 self)
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: We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that `(x; q; a) ¸ 2x=OE(q) for each a 62 H and `(x; q; a) = o(x=OE(q)), otherwise. From here, we deduce that if the second case holds at all, then it holds only for the multiples of some fixed integer q 0 ? 1. Actually, even if the Prime Number Theorem for arithmetic progressions, modulo q, does hold, these methods allow us to deduce the behaviour of a possible `Siegel zero' from Selberg's formula. We also propose a new method for determining explicit upper and lower bounds on `(x; q; a), which uses only elementary number theoretic computations. 1. Introduction. Define `(x) = P px log p, where p only denot...
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.
An explicit zerofree region for the Dirichlet Lfunctions, ArXiv : math.NT/0510570
"... Abstract. Let Lq(s) be the product of Dirichlet Lfunctions modulo q. Then Lq(s) has at most one zero in the region ..."
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Abstract. Let Lq(s) be the product of Dirichlet Lfunctions modulo q. Then Lq(s) has at most one zero in the region
Values of the Euler φfunction not divisible by a given odd prime, and the distribution of . . .
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AN IMPROVED UPPER BOUND FOR THE ARGUMENT OF THE RIEMANN ZETAFUNCTION ON THE CRITICAL LINE
"... Abstract. This paper concerns the function S(t), the argument of the Riemann zetafunction along the critical line. Improving on the method of Backlund, and taking into account the refinements of Rosser and McCurley it is proved that for sufficiently large t S(t)  ≤ 0.1013 log t. Theorem 2 makes ..."
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Abstract. This paper concerns the function S(t), the argument of the Riemann zetafunction along the critical line. Improving on the method of Backlund, and taking into account the refinements of Rosser and McCurley it is proved that for sufficiently large t S(t)  ≤ 0.1013 log t. Theorem 2 makes the above result explicit, viz. it enables one to select values of a and b such that, for t> t0, S(t)  ≤ a + b log t. 1.
SHORT EFFECTIVE INTERVALS CONTAINING PRIMES IN ARITHMETIC PROGRESSIONS AND THE SEVEN CUBES PROBLEM
"... Abstract. For any ɛ>0 and any nonexceptional modulus q ≥ 3, we prove that, for x large enough (x ≥ αɛ log 2 q), the interval [ e x,e x+ɛ] contains a prime p in any of the arithmetic progressions modulo q. We apply this result to establish that every integer n larger than exp(71 000) is a sum of sev ..."
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Abstract. For any ɛ>0 and any nonexceptional modulus q ≥ 3, we prove that, for x large enough (x ≥ αɛ log 2 q), the interval [ e x,e x+ɛ] contains a prime p in any of the arithmetic progressions modulo q. We apply this result to establish that every integer n larger than exp(71 000) is a sum of seven cubes. 1.
A modest improvement on the function S(T)
, 2010
"... This paper concerns the function S(T), the argument of the Riemann zetafunction. Improving on the method of Backlund, and taking into account the refinements of Rosser and McCurley it is hereunder proved that for sufficiently large T S(T)  ≤ 0.1013 log T. Theorem 2 makes the above result explici ..."
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This paper concerns the function S(T), the argument of the Riemann zetafunction. Improving on the method of Backlund, and taking into account the refinements of Rosser and McCurley it is hereunder proved that for sufficiently large T S(T)  ≤ 0.1013 log T. Theorem 2 makes the above result explicit, viz. it enables one to select values of a and b such that, for T> T0, S(T)  ≤ a + b log T. 1