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34
Computational Strategies for the Riemann Zeta Function
 Journal of Computational and Applied Mathematics
, 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 46 (9 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 13 (1 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 every x ..."
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Cited by 13 (6 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.
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 th ..."
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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 Dirichlet Lfunctions off the critical line
 Duke Math. J
"... ABSTRACT. We show, for any q � 3 and distinct reduced residues a,b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet Lfunctions lying off the critical line implies that π(x;q,a) < π(x;q,b) for a set of real x of asymptotic density 1. 1 ..."
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Cited by 2 (0 self)
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ABSTRACT. We show, for any q � 3 and distinct reduced residues a,b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet Lfunctions lying off the critical line implies that π(x;q,a) < π(x;q,b) for a set of real x of asymptotic density 1. 1
Chebyshev's Bias
, 1994
"... this paper we take a somewhat different point of view in our attempt to analyze Chebyshev's phenomenon and its generalizations, which we call "Chebyshev's bias". Our purpose has been to examine these issues both theoretically and numerically and, in particular, to give numerical values to these bias ..."
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Cited by 1 (0 self)
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this paper we take a somewhat different point of view in our attempt to analyze Chebyshev's phenomenon and its generalizations, which we call "Chebyshev's bias". Our purpose has been to examine these issues both theoretically and numerically and, in particular, to give numerical values to these biases. Let a 1 ; a 2 ; : : : ; a r 2 A q be distinct, and define
Arithmetic Properties of Class Numbers of Imaginary Quadratic Fields
, 2006
"... Under the assumption of the wellknown heuristics of Cohen and Lenstra (and the new extensions we propose) we give proofs of several new properties of class numbers of imaginary quadratic number fields, including theorems on smoothness and normality of their divisors. Some applications in cryptograp ..."
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Cited by 1 (0 self)
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Under the assumption of the wellknown heuristics of Cohen and Lenstra (and the new extensions we propose) we give proofs of several new properties of class numbers of imaginary quadratic number fields, including theorems on smoothness and normality of their divisors. Some applications in cryptography are also discussed. 1
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 1 (1 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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Cited by 1 (0 self)
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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.
Primes in almost all short intervals and the distribution of the zeros of the Riemann zetafunction
"... We study the relations between the distribution of the zeros of the Riemann zetafunction and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)#(xy) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available ..."
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Cited by 1 (1 self)
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We study the relations between the distribution of the zeros of the Riemann zetafunction and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)#(xy) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available zerofree regions for the Riemann zetafunction, and also on the strength of density bounds for the zeros themselves. We also study implications in the opposite direction: assuming that an asymptotic formula like the above is valid for almost all x in a given range of values for y, we find zerofree regions or density bounds.