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13
Higher correlations of divisor sums related to primes, II: Variations of . . .
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
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Cited by 28 (6 self)
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet Lfunctions, we obtain an Ω±result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ωresults for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.
Li Coefficients for Automorphic LFunctions
, 2004
"... XianJin Li gave a criterion for the Riemann hypothesis in terms of the positivity of the set of coefficients λn = ∑ ( ρ 1 − 1 − 1) n ..."
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Cited by 5 (2 self)
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XianJin Li gave a criterion for the Riemann hypothesis in terms of the positivity of the set of coefficients λn = ∑ ( ρ 1 − 1 − 1) n
On Fourier and Zeta(s)
, 2002
"... This is the final version for FORUM MATHEMATICUM; October 2002 We study some of the interactions between the Fourier Transform and the Riemann zeta ..."
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Cited by 1 (1 self)
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This is the final version for FORUM MATHEMATICUM; October 2002 We study some of the interactions between the Fourier Transform and the Riemann zeta
AVERAGE PRIMEPAIR COUNTING FORMULA
, 2009
"... Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) =π2r(x)−2C2r li2(x) th ..."
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Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) =π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x)−li(x). However, there is a heuristic approximate formula for averages of the remainders ω2r(x) which is supported by numerical results. 1.
Article electronically published on September 25, 2009 AVERAGE PRIMEPAIR COUNTING FORMULA
"... Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) =π2r(x)−2C2r li2(x) th ..."
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Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) =π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x)−li(x). However, there is a heuristic approximate formula for averages of the remainders ω2r(x) which is supported by numerical results. 1.
Fast methods to compute the Riemann zeta function
, 2008
"... The Riemann zeta function on the critical line can be computed using a straightforward application of the RiemannSiegel formula, Schönhage’s method, or HeathBrown’s method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new fast and p ..."
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The Riemann zeta function on the critical line can be computed using a straightforward application of the RiemannSiegel formula, Schönhage’s method, or HeathBrown’s method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author’s algorithm to compute quadratic exponential sums. Its complexity has exponent 1/3. The third method employs an algorithm, developed in this paper, to compute cubic exponential sums. Its complexity has exponent 4/13 (approximately, 0.307). 1
LOWER BOUND FOR THE REMAINDER IN THE PRIMEPAIR CONJECTURE
, 806
"... Abstract. Taking r> 0 let π2r(x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. A heuristic argument indicates that the remainder e2r(x) in this approximation canno ..."
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Abstract. Taking r> 0 let π2r(x) denote the number of prime pairs (p, p + 2r) with p ≤ x. The primepair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. A heuristic argument indicates that the remainder e2r(x) in this approximation cannot be of lower order than x β, where β is the supremum of the real parts of zeta’s zeros. The argument also suggests an approximation for π2r(x) similar to one of Riemann for π(x). 1.
THE FIRST MOMENT OF QUADRATIC DIRICHLET LFUNCTIONS
, 804
"... Abstract. We obtain an asymptotic formula for the smoothly weighted first moment of primitive quadratic Dirichlet Lfunctions at the central point, with an error term that is “squareroot ” of the main term. Our approach uses a recursive technique that feeds the result back into itself, successively ..."
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Abstract. We obtain an asymptotic formula for the smoothly weighted first moment of primitive quadratic Dirichlet Lfunctions at the central point, with an error term that is “squareroot ” of the main term. Our approach uses a recursive technique that feeds the result back into itself, successively improving the error term. 1.
THE SECOND MOMENT OF QUADRATIC TWISTS OF MODULAR LFUNCTIONS
, 907
"... The family of quadratic twists of a modular form has received much attention in recent years. Motivated by the BirchSwinnertonDyer conjectures, we seek an understanding of the central values of the associated Lfunctions, and while this question has been investigated extensively, much remains unkn ..."
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The family of quadratic twists of a modular form has received much attention in recent years. Motivated by the BirchSwinnertonDyer conjectures, we seek an understanding of the central values of the associated Lfunctions, and while this question has been investigated extensively, much remains unknown. One important theme in this area concerns the moments