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".

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 so-called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants. 1.

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.

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

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

Under the assumption of the well-known 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

We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)-#(x-y) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available zero-free regions for the Riemann zeta-function, 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 zero-free regions or density bounds.

It is unknown whether there is an infinite number of twin primes, which are primes differing by two. Another question for which an answer is known for primes, but not for the twin primes, is how they are distributed across residue classes. There are generalized arguments that describe when there are more primes that have one remainder when divided by a particular number than another remainder. For example there are more primes 1 mod 10 than 7 mod 10, that is, end in a 1 than a 7 when written in base 10. For twin primes there are similar questions. Are there more twin primes that end in 1 and 3 or twin primes that end in 7 and 9? We develop an experimental method

ABSTRACT. Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers � x that are in a given arithmetic progression modulo q and the product of two primes. The two assumptions are (i) the Extended Riemann Hypothesis for Dirichlet L-functions modulo q, and (ii) that the imaginary parts of the nontrivial zeros of these L-functions are linearly independent over the rationals. Our results are analogs of similar results proved for primes in arithmetic progressions by Rubinstein and Sarnak. 1.

Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The prime-pair 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.