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We gratefully acknowledge support from the Simons Foundation and member institutions. Subjects: Number Theory math. Long shift ranges. Comments: 80 pages, no figures. Journal-ref: Proc. Journal-ref: Chebyshevskii Sbornik, volume 19, no. AP ; Probability math.
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Authors: Florin P. Comments: 42 pages; section 5 re-written following referee report. Divisor correlations in short ranges. Comments: 41 pages; minor revision, taking into account the referee's comments, to appear in Ann.
See equation 9 and the footnote below. Comments: 26 pages; corrected a small mistake in Ramare's identity. See equation 3. Journal-ref: Algebra Number Theory 9 Analytic Number Theory Donald J. Analytic Number Theory Y. Analytic Number Theory , Y. Number Theory Books, Grabner editors , Springer I welcome your requests and suggestions for library acquisitions.
Preprints and Papers by Christian Elsholtz. Number Theory :: BookNavigator. Chi QS, Wang H. Beitrage zur … Read more. Campus Box One Brookings Dr. A more correct description of his results would be that they make effective earlier work of Dahlquist; see.
Dahlquist, On the analytic continuation of Eulerian products. Selfridge , article A14, 19 pages electronic. Abstract We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples.
For example, all even perfect numbers are prime-perfect. Monthly , ; abridged errata in , Abstract Since ancient times, a natural number has been called perfect if it equals the sum of its proper divisors; e. We show how this result, and many like it, follow from embedding the natural numbers in the supernatural numbers and imposing an appropriate topology on the latter; the notion of sequential compactness plays a starring role. Number Theory 9 , Algebra Number Theory 6 , no. Here we prove a result complementary to that of Ford et al. Similar results with the average taken over all quadratic fields were recently proved by the second author.
Finite Fields and their Applications 22 , Their conjecture was established in all but finitely many cases by Wan, and later resolved in full in work of Ham and Mullen. In this note, we present a new, explicit estimate for the number of irreducibles with several preassigned coefficients.
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Debrecen 82 , Ramanujan J. This confirms a conjecture of Pomerance. This strengthens a recent theorem of Olofsson. In this paper, we prove two statistical theorems about these classes of numbers. Ruzsa, and V. Here we prove two significantly stronger versions of this uncertainty principle. In , Davenport showed that all three of these sets make up a well-defined proportion of the positive integers.
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Number Theory 10 , Abstract Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. Mathematika 60 , In , Bessel-Hagen asked whether or not the set of abundant numbers possesses an asymptotic density. Four years later, Davenport answered Bessel-Hagen's question in the affirmative. Abstract We show that almost all squares are missing from the range of Euler's totient function.
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Monthly , For each of these sequences, he asked whether every prime eventually appears. Recently, Booker showed that the second sequence omits infinitely many primes. We give a completely elementary proof of Booker's result, suitable for presentation in a first course in number theory. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example:. Our estimate also allows us to recover the upper bounds of Vinogradov--Linnik, Elliott, and the present author on the least split-completely prime in an abelian number field.
Algebra Number Theory 8 , But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. We inject Hooley's analysis into the Maynard--Tao work on bounded gaps between primes. Illinois J. Abstract In a paper, Euler gave the first proof that the sum of the reciprocals of the prime numbers diverges. That paper can be considered as the founding document of analytic number theory, and its key innovation so-called Euler products are now ubiquitous in the field.
In this note, we probe Euler's claim there that "the sum of the reciprocals of the prime numbers" is "as the logarithm" of the sum of the harmonic series. Euler's argument for this assertion falls far short of modern standards of rigor. We describe a simple proof that the sum of the reciprocals of all such primes diverges. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem.
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-  Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao;
Paris , We also show how to reach the same conclusion for several other commonly occurring arithmetic functions. Number Theory 11 , Every amicable pair is harmonious, but there are many others.
The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length. Math 21 , Abstract We revisit Besicovitch's paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper's influence, the results he inspired are not strong enough to reprove his original result.
Rassias and C. Pomerance, eds. We describe an application of the second result to the theory of L. Dickson's amicable tuples, which are a generalization of the ancient notion of an amicable pair. Number Theory 5 , Cambridge Philos Soc. The proof uses number field analogues of the Brun--Titchmarsh and Bombieri--Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions. Notices , no. This sharpens previous estimates of Friedlander and Iwaniec.
Forum Mathematicum 28 , This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin's conjecture. Assume GRH. Rocky Mountain J. Abstract Tao has shown that in any fixed base, a positive proportion of prime numbers cannot have any digit changed and remain prime. In other words, most primes are "digitally delicate". We strengthen this result in a manner suggested by Tao: A positive proportion of primes become composite under any change of a single digit and any insertion a fixed number of arbitrary digits at the beginning or end.
To establish these analytic results we need to extend some prior algebraic results. Notices , We then present an application to a problem in arithmetic statistics. The proof uses the fundamental lemma of the sieve, Norton's refinement of the Burgess bounds, and a result of Tenenbaum on the distribution of smooth numbers satisfying a coprimality condition.
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Japan Acad. A Math. This generalizes a result of Cho. In this note, we study the frequency of champion and trailing primes for CM elliptic curves. Horizons of Fractal Geometry and Complex Dimensions. In celebration of the 60th birthday of Michel Lapidus. Niemeyer, E. Pearse, J. Rock, T. Samuel, eds. Abstract Consider the Laplacian operator on a bounded open domain in Euclidean space with Dirichlet boundary conditions.
The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant.
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