Prime Number Theory Research Articles

Book Review: Introduction to modern prime number theory

Book Review: Introduction to modern prime number theory

A Summation Formula in the Theory of Prime Numbers

A Summation Formula in the Theory of Prime Numbers

SOME FOURIER TRANSFORMS IN PRIME-NUMBER THEORY

Journal Article SOME FOURIER TRANSFORMS IN PRIME-NUMBER THEORY Get access A. P. GUINAND A. P. GUINAND Oxford Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 53–64, https://doi.org/10.1093/qmath/os-18.1.53 Published: 01 January 1947 Article history Received: 15 August 1946 Published: 01 January 1947

The Theory of Measure in Arithmetical Semi-Groups

THIS monograph, the title of which is somewhat abstruse, deals in fact with certain aspects of prime-number theory. Though self-contained in presentation, it is a continuation of the author's "Eratosthenian Averages"(Baltimore, 1943). A knowledge of analytical prime-number theory and of Tauberian theorems is assumed, and even readers possessing this knowledge will probably not find the book easy reading. The greater part of it is concerned with the properties of the set of all integers that are divisible only by the primes of a given sequence, and in particular the various senses in which the set may have a 'density'.

Additive prime number theory in real quadratic fields

Additive prime number theory in real quadratic fields

LXXIX.Application of the operational or symbolic calculus to the theory of prime numbers

LXXIX.<i>Application of the operational or symbolic calculus to the theory of prime numbers</i>

The zeros of the Riemann zeta-function

1―It is well known that the distribution of the zeroes of the Riemann zeta-function ζ( s ) = ∞ Σ n=1 1/ n 8 ( s = σ + it ) plays a fundamental part in the theory of prime numbers. It was conjectured by Riemann that all the complex zeroes of ζ( s ) lie on the line σ = 1/2, but this hypothesis has never been proved or disproved. It is therefore natural to enquiry how far the hypothesis is supported by numerical calculations. The most extensive calculations of this kind have been undertaken by Gram, Backlund, and Hutchinson. The final result obtained by Hutchinson is that ζ( s ) has 138 zeroes on σ = 1/2 between t = 0 and t = 300, and no other zeroes between these values of t .

News and Views

HIS MAJESTY THE KING has approved of the following awards this year by the president and council of the Royal Society in respect of the two Royal medals: A Royal medal to Prof. J. E. Littlewood, for his work on mathematical analysis and the theory of prime numbers. A Royal medal to Prof. R. Muir, for his contributions to the science of immunology. The following awards have also been made by the president and council: The Copley medal to Prof. Max Planck, of the University of Berlin, for his contributions to theoretical physics, and especially as the originator of the quantum theory. The Davy medal to Prof. G. N. Lewis, of the University of California, for his contributions to classical thermodynamics and the theory of valency. The Hughes medal to Prof. Hans Geiger, of the University of Kiel, for his invention and development of methods of counting alpha and beta particles.

A Functional Equation from the Theory of the Riemann ζ(s)-Function

In. the following brief paper a functional equality is deduced by means of which the well-known Riemann formula from the analytic theory of prime numbers may be obtained. The method which has been followed in this connection may be followed -also, without any difficulty, in the case of more general C-functions. In Acta Miathematica vol. 25, p. 166, Mellin establishes the formula a+ivo (1 ) log (l+7 2ijbsi (0 < a c ), a-ioo

Some asymptotic expressions in the theory of numbers

While formerly the research of asymptotic expressions in the theory of numbers was largely confined to the approximate determination of the summatory function (or the mean value) of a given arithmetic function, recent progress in the theory of prime numbers has opened a new field for asymptotic investigations, viz., the research of upper and lower limits of an arithmetic function for large values of its argument. The first result in this line was obtained by LANDAU:t