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52
Computational strategies for the Riemann zeta function
, 2000
"... We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of th ..."
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Cited by 67 (11 self)
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We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call “value recycling”.
Prime Number Races
 Amer. Math. Monthly
"... 1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that ..."
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Cited by 23 (2 self)
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1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking
Chebyshev’s bias for composite numbers with restricted prime divisors
 Math. Comp
, 2005
"... Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for ever ..."
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Cited by 16 (7 self)
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Abstract. Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n)>π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. Ifπ(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) denotes the number of integers n ≤ x such that every prime divisor p of n satisfies p ≡ a(mod d). In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, N(x;4, 3) ≥ N(x;4, 1) for every x. In the process we express the socalled second order LandauRamanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants. 1.
The primecounting function and its analytic approximations  π(x) and its approximations
, 2008
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A NEW BOUND FOR THE SMALLEST x WITH π(x)> li(x)
, 1999
"... Abstract. Let π(x) denote the number of primes ≤ x and let li(x) denotethe usual integral logarithm of x. We prove that there are at least 10153 integer values of x in the vicinity of 1.39822 × 10316 with π(x)> li(x). This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more ..."
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Cited by 9 (0 self)
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Abstract. Let π(x) denote the number of primes ≤ x and let li(x) denotethe usual integral logarithm of x. We prove that there are at least 10153 integer values of x in the vicinity of 1.39822 × 10316 with π(x)> li(x). This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of π(x) − li(x) in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of 1.617 × 109608,whereπ(x) appears to exceed li(x) bymore than.18x 1 2 / log x. The plots strongly suggest, although upper bounds derived to date for li(x) − π(x) are not sufficient for a proof, that π(x) exceeds li(x) for at least 10311 integers in the vicinity of 1.398 × 10316. If it is possible to improve our bound for π(x) − li(x) by finding a sign change before 10316,our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of li(x) − π(x) and find that as x departs from the region in the vicinity of 1.62 × 109608, the density is 1 − 2.7 × 10−7 =.99999973, and that it varies from this by no more than 9 × 10−8 over the next 1030000 integers. This should be compared to Rubinstein and Sarnak.
THE PRIME NUMBER RACE AND ZEROS OF LFUNCTIONS OFF THE CRITICAL LINE
"... We examine the effects of certain hypothetical configurations of zeros of Dirichlet Lfunctions lying off the critical line on the distribution of primes in arithmetic progressions. 1. ..."
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Cited by 5 (2 self)
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We examine the effects of certain hypothetical configurations of zeros of Dirichlet Lfunctions lying off the critical line on the distribution of primes in arithmetic progressions. 1.
Riemann and his zeta function
, 2005
"... An exposition is given, partly historical and partly mathematical, of the Riemann zeta function ζ(s) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count nontrivial zeros of ζ(s). Relevance of these investigations to the theory of ..."
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An exposition is given, partly historical and partly mathematical, of the Riemann zeta function ζ(s) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count nontrivial zeros of ζ(s). Relevance of these investigations to the theory of the distribution of prime numbers is discussed.
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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Cited by 4 (0 self)
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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.
Counting numbers in multiplicative sets: Landau versus Ramanujan
 Math. Newsl
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