Richard H. Hudson

A new bound for the smallest x with p(x) li (x)

Let π(x) denote the number of primes 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 x 10 9608 , where π(x) appears to exceed li(x) by more 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 10 311 integers in the vicinity of 1.398 x 10 316 . If it is possible to improve our bound for π(x)-li(x) by finding a sign change before 10 316 , 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 x 10 9608 , the density is 1 - 2.7 x 10 -7 =.99999973, and that it varies from this by no more than 9 x 10 -8 over the next 10 30000 integers. This should be compared to Rubinstein and Sarnak.

The segmented sieve of eratosthenes and primes in arithmetic progressions to 1012

The sieve of Eratosthenes, a well known tool for finding primes, is presented in several algorithmic forms. The algorithms are analyzed, with theoretical and actual computation times given. The authors use the sieve in a refined form (the “dual sieve”) to find the distribution of primes in twenty arithmetic progressions to 1012. Tables of values are included.

Binomial coefficients and Jacobi sums

ABSTRACT. Throughout this paper e denotes an integer > 3 and p a prime-1 (mod e). With f defined by p = ef + 1 and for integers r and s satisfying 1 s s < r < e-1, certain binomial coefficients (

Determination of all imaginary cyclic quartic fields with class number 2

) have been determined in terms of the parameters in various binary and quaternary quadratic forms by, for example, Gauss [13], Jacobi [19,20], Stern [37-40], Lehmer [23] and Whiteman [42,45,46]. In §2 we determine for each e the exact number of binomial coefficients (

Calculation of the class numbers of imaginary cyclic quartic fields

) not trivially congruent to one another by elementary properties of number theory and call these representative binomial coefficients. A representative binomial coefEcient is said to be of order e if and only if ( r, s ) = 1. In §§3-4, we show how the Davenport-Hasse relation [71, in a form given by Yamamoto [50], leads to determinations of n(p-i)/t in terms of binomial coefScients modulo p = ef + 1 = mnf + 1. These results are of some interest in themselves and are used extensively in later sections of the paper. Making use of Theorem 5.1 relating Jacobi sums and binomial coefEcients, which was Srst obtained in a slightly different form by Whiteman [45], we systematically investigate in §§6-21 all representative binomial coefEcients of orders e = 3,4,6,7,8,9,11, 12, 14, 15, 16, 20 and 24, which we are able to determine explicitly in terms of the parameters in well-known binary quadratic forms, and all representative binomial coefEcients of orders e = 5,10, 13, 15, 16 and 20, which we are able to explicitly determine in terms of quaternary quadratic decompositions of 16p given by Dickson [9], Zee [51] and Guidici, Muskat and Robinson [14]. Some of these results have been obtained by previous authors and many new ones are included. For e = 7 and 14 we are unable to explicitly determine representative binomial coefEcients in terms of the six variable quadratic decomposition of 72p given by Dickson [9] for reasons given in §10, but we are able to express these binomial coefficients in terms of the parameter xl in this system in analogy to a recent result of Rajwade [34]. Finally, although a relatively rare occurrence for small e, it is possible for representative binomial coefEcients of order e to be congruent to one another (mod p). Representative binomial coefficients which are congruent to + 1 times at least one other representative for all p = ef + 1 are called Cauchy-Whiteman type binomial coefEcients for reasons given in [171 and §21. All congruences between such binomial coefEcients are carefully exarrAined and proved (with the sign ambiguity removed in each case) for all values of e considered. When e = 24 there are 48 representative binomial coefEcients, including those of lower order, and it is shown in §21 that an astonishing 43 of these are Cauchy-Whiteman type binomial coefScients. It is of particular interest that the sign ambiguity in many of these congruences does not arise from any expression of the form n(PI)/m in contrast to the case for all e < 24.

On the fluctuations of Littlewood for primes of the form 4̸=1

It is proved that there are exactly 8 imaginary cyclic quartic fields with class number 2.

A formula for the exact number of primes below a given bound in any arithmetic progression

Any imaginary cyclic quartic field can be expressed uniquely in the form K = Q(\JA(D + b/d) ), where A is squarefree, odd and negative, D = B2 + C2 is squarefree, B > 0, C > 0, and (A,D)= 1. Explicit formulae for the discriminant and conductor of K are given in terms of A, B, C, D. The calculation of tables of the class numbers h(K) of such fields K is described. Let Q denote the field of rational numbers and let K be a cyclic extension of Q of degree 4. The unique quadratic subfield of K is denoted by k. The class number of K (resp. k) is denoted by h(K) (resp. h(k)). The conductor of K is denoted by / = /( K ). In the case of real cyclic quartic fields K, Gras (3) has given a table of the values of h(K) for all such fields with / < 4000. Recently, the authors have carried out the calculation of the class numbers of imaginary cyclic quartic fields (4). In this note we give a brief description of the computation of the tables given in (4). The following explicit representation of a cyclic quartic field is proved in (4,

Zeroes of Dirichlet L -functions and irregularities in the distribution of primes

Let rb,c(x) denote the number of primes 6 x which are c (mod b). Among the first 950,000,000 integers there are only a few thousand integers n with r4,3(n) < or4jl(n). The authors find three new widely spaced regions containing hundreds of millions of such integers; the density of these integers and the spacing of the regions is of some importance because of their intimate connection with the truth or falsity of the analogue of the Riemann hypothesis for L(s). The discovery that the majority of all integers n less than 2 X 1010 with r4,3(n) < ir4,1(n) are the 410,000,000 (consecutive) integers lying between 18,540,000,000 and 18,950,000,000 is a major surprise; results are carefully corroborated and some of the implications are discussed.

Details of the first region of integers with _{3,2}()<_{3,1}()

The formula of [E.] Meissel [ Math. Ann . 2 (1870), 636–642] is generalized to arbitrary arithmetic progressions. Meissels formula is applicable not only to computation of π( x ) for large x (recently x = 10 13 ), but also is a sieve technique (see MR36#2548), useful for studying the subtle effect of primes less then or equal to x 1/2 on the behavior of primes less than or equal to x . The same is true of the generalized Meissel, with the added advantage that the behavior of primes less than or equal to x can be studied in arbitrary progressions.

On the first occurrence of certain patterns of quadratic residues and non-residues

Seven widely spaced regions of integers with π4,3(x) < π4,1(x) have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes it possible to compute the entire distribution of π4,3(x) - π4,1(x) including the sign change (axis crossing) regions, in time linear in x, using zeroes of L(s,x),x the nonprincipal character modulo 4, generously provided to us by Robert Rumely. The accuracy with which the zeroes duplicate the distribution (Figure 1) is very satisfying. The program discovers all known axis crossing regions and finds probable regions up to 10 1000 Our result is applicable to a wide variety of problems in comparative prime number theory. For example, our theorem makes it possible in a few minutes of computer time to compute and plot a characteristic sample of the difference li(x)-π(x) with fine resolution out to and beyond the region in the vicinity of 6.658 x 10 370 discovered by te Riele. This region will be analyzed elsewhere in conjunction with a proof that there is an earlier sign change in the vicinity of 1.39822 × 10 316 .