Abstract: : The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before They give improved methods for using this and a recent determination that 3,500,000 zeros lie on the critical line to develop better bounds for functions of primes

Abstract: / = lim inf ̂ 11-tl . »->» log pn The purpose of this paper is to combine the methods used in two earlier papers1 in order to prove the following theorem. Theorem. (1) / = c(l + 40)/5, where c< 42/43. The precise definition of c is given by (18) and (8) below. The theorem is an improvement on the result /^(14-40)/5 obtained in II. It was shown in III that (2) I = d = 0.966 < 57/59. The numbers c and d are connected by the relation 1 d = (3 42\"2)(1 c)/4 > 1 c, so that (1) is an improvement on (2) only if © is not too close to unity, in fact, if 0< 0.986 • • • . In particular, if the \"grand Riemann hypothesis\" is true, that is, if 0 = 1/2, we have (using the value of c given by (18)) / ^ 3c/5 < 109/186 = 0.58602 2. Notation. Since (2) is sharper than (1) for 0 = 1, we shall assume that 0<1, and write (3) a = (1 + 4©)/5. Let N be a large positive integer, and define (4) X = A^-'Oog Af)-»«+T)

Abstract: In this paper we shall prove that for a large positive number x there exist at least two integers n in the interval x-x~(1/2)n≤x having at most two prime factors.

Abstract: In this paper, we give a modified proof of Chen's theorem "every sufficiently large even inte- ger is a sum of a prime and a product of at most 2 primes".

Abstract: In this note we give a simple formula for the nth prime number. Let pn denote the nth prime number (p 1=2, p 2 = 3, etc.). We shall show that p n is given by the following formula.

Abstract: Let be a prime number, an algebraic number field containing a primitive th root of unity, a finite set of valuations of containing all prime divisors of , and the maximal -extension of unramified outside .The paper studies local extensions for , and the corresponding decomposition subgroups . It is proved that in almost all cases coincides with the maximal -extension of ; in particular, this holds if . Also, a series of results is obtained on the relative location of the various in , and the group of universal norms from the group of -units of to the group of -units of is computed.Bibliography: 7 items.

Abstract: It is shown that complete sets of $( u - 1)/2$ by $ u$ and $( u + 1)/2$ by $ u$ multistage balanced Youden designs of type I and II can be constructed if $ u$, the number of treatments, is a prime power of the form $4\lambda + 3$. It is also proved that the existence of a difference set with certain properties implies the existence of a 2-stage balanced Youden design. The usefulness of this latter result is demonstrated for those experiments where the number of treatments is not a prime power.

Abstract: The "/?-adic analytic number theory" alluded to in the title of my article is in a very beginning state: [4], [6], [2]. In different contexts, and from different points of view, p-adic analytic number theory has been the subject of much recent work : "the /7-adic analytic number theory of totally real number fields" has been developed by Serre [10], using work of Siegel, and more recently by Katz, and Deligne-Ribet; "of quadratic imaginary number fields" : by Katz, and Manin; "of modular forms of weight k ^ 2 for the full modular group" : by Manin [3]; "of Eichler cohomology classes associated to certain arithmetic groups" : being presently worked on by V. Miller. One exciting aspect of this emerging theory is its sheer difficulty : for example, no matter which elliptic curve E/Q you choose (e.g., y + y = x + x)9 its /?-adic analytic number theory is hard to get to know intimately for most primes p9 either conceptually or computationally. Nevertheless, for the jacobian of the modular curve XQ(N)/Q, there are certain special primes where things are under better control, and for which a more precise picture is beginning to come into view. One obtains a number of by-products of this picture which are of independent diophantine interest. Notably, as we shall discuss below, one can prove Mordell's conjecture for the curves XQ(N) for prime N over Q. For general prime numbers N the "Mordell conjecture" is proven in a bleakly indeterminate form; Ogg and I have been working with (and sharpening) the result, however, and have obtained an actu-

Abstract: The maxima and minima of 〈L(x)〉−π(x), 〈R(x)〉−π(x), and 〈L2(x)〉−π2(x) in various intervals up to x = 8×1010 are tabulated. Here π(x) and π2(x) are respectively the number of primes and twin primes not exceeding x, L(x) is the logarithmic integral, R(x) is Riemann’s approximation to π(x), and L2(x) is the Hardy-Littlewood approximation to π2(x). The computation of the sum of inverses of twin primes less than 8×1010 gives a probable value 1.9021604±5×10−7 for Brun’s constant. Comments Only the Abstract is given here. The full paper appeared as [1]. For a more recent evaluation of Brun’s constant, which incidentally resulted in the discovery of a bug in the Pentium floating- point divide, see [3]. Errata. On page 49, three lines from the bottom, “17” should be replaced by “16”, and “900” by “960”. On the same page, six lines from the bottom, “17” should be replaced by “16”. References [1] R. P. Brent, “Irregularities in the distribution of primes and twin primes”, Mathematics of Computation (Derrick H. Lehmer special issue) 29 (1975), 43–56. MR 50#1791, 51#5522; Zbl 295.10002. Errata: ibid 30 (1976), 198. MR 53#302. See also “Tables concerning irregularities in the distribution of primes and twin primes”, UMT 4, ibid 29 (1975), 331; and “Tables concerning irregularities in the distribution of primes and twin primes to 10”, UMT 21, ibid 30 (1976), 379. rpb024. [2] R. Sherman Lehman, “On the difference π(x)− li(x)”, Acta Arithmetica 11 (1966), 397–410. MR 34#2546. [3] T. R. Nicely, “Enumeration to 10 of the Twin Primes and Brun’s Constant”, Virginia Journal of Science 46 (1995), 195–204. For a review see review04.dvi.gz . Computer Centre, Australian National University, Canberra, Australia 1991 Mathematics Subject Classification. Primary 11-04, 11Y60, 11Y70; Secondary 11A41, 65A05, 65B05.

Abstract: Let x, y, Q denote large real numbers, with x > y > Q. The object of this paper is to prove the following resultfor arbitrary A > 0, with a larger value of Q than hitherto. The symbol denotes as usual the suppression of an absolute constant, π(N; q, a) denotes the number of prime numbers up to N which are congruent to a (mod q), and ϕ(q) denotes Euler's function.