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No description available.

Identified as one of the 7 Millennium Problems, the Riemann zeta hypothesis has successfully evaded mathematicians for over 100 years. Simply stated, Riemann conjectured that all of the nontrivial zeroes of his zeta function have real part equal to 1/2. This thesis attempts to explore the theory behind Riemann’s zeta function by first starting with Euler’s zeta series and building up to Riemann’s function. Along the way we will develop the math required to handle this theory in hopes that by the end the reader will have immersed themselves enough to pursue their own exploration and research into this fascinating subject.

Thesis (M. Sc.)(Mathematics)--University of Pretoria, 2006. / Includes bibliographical references. Available on the Internet via the World Wide Web.

No description available.

The aim of this thesis is to examine the connection between the Riemannhypothesis and the distribution of prime numbers. We first derive theanalytic continuation of the zeta function and prove some of its propertiesusing a functional equation. Results from complex analysis such asJensen’s formula and Hadamard factorization are introduced to facilitatea deeper investigation of the zeros of the zeta function. Subsequently, therelation between these zeros and the asymptotic distribution of primesis rendered explicit: they determine the error term when the prime-counting function π(x) is approximated by the logarithmic integral li(x).We show that this absolute error is O(x exp(−c√log x) ) and that the Riemannhypothesis implies the significantly improved upper bound O(√x log x).

We follow a research paper that J. Elstrodt published in 1998 to prove the Prime Number Theorem for arithmetic progressions. We will review basic results from Dirichlet characters and L-functions. Furthermore, we establish a weak version of the Wiener-Ikehara Tauberian Theorem, which is an essential tool for the proof of our main result.

This thesis is an educational text for high school students. It aims to teach them how to solve typical problems concerning divisibility found in mathematical correspondence seminars and mathematical olympiad. Basic notions from the theory of divisibility are recalled (e.g. prime numbers, divisors, multiples). Criteria of divisibility by 2 to 20 are introduced, as well as diophantine equations and practical applications of prime numbers in real life. One whole chapter is dedicated to problems and exercises. Powered by TCPDF (www.tcpdf.org)

Η παρούσα διπλωματική εργασία έχει στόχο τη μελέτη της συνάρτησης Ζήτα του Riemann μέσω της Μιγαδικής ανάλυσης δηλαδή ως μία επέκταση αυτής από την ευθεία των πραγματικών αριθμών στο μιγαδικό επίπεδο. Η σύνδεση της συνάρτησης αυτής με τους πρώτους αριθμούς, η διάσημη υπόθεση Riemann, η συναρτησιακή εξίσωση, η αναλυτικότητά της εκτός σημείου είναι μερικά αποτελέσματα της μελέτης. Το αρχικό βήμα της εργασίας πριν από την μελέτη της συνάρτησης Ζήτα είναι μια εκτενής αναφορά στην συνάρτηση Γάμμα ως επεκτεινόμενη στο Μιγαδικό επίπεδο, την αναλυτικότητά της και κάποιων βασικών ιδιοτήτων της. Κάτι τέτοιο ήταν αναγκαίο, διότι η συνάρτηση Γάμμα αποτελεί ένα εργαλείο για τη μελέτη της συνάρτησης Ζήτα. / --

<div><div><div><p>It is well-known that the sum of reciprocals of twin primes converges or is a finite sum.</p><p>In the same spirit, Samuel Wagstaff proved in 2021 that the sum of reciprocals of primes p</p><p>such that ap + b is prime also converges or is a finite sum for any a, b where gcd(a, b) = 1</p><p>and 2 | ab. Wagstaff gave upper and lower bounds in the case that ab is a power of 2. Here,</p><p>we expand on his work and allow any a, b satisfying gcd(a, b) = 1 and 2 | ab. Let Πa,b be the</p><p>product of p−1 over the odd primes p dividing ab. We show that the upper bound of these p−2</p><p>sums is Πa,b times the upper bound found by Wagstaff and provide evidence as to why we cannot hope to do better than this. We also give several examples for specific pairs (a, b).</p><p><br></p><p>Next, we turn our attention to elliptic Carmichael numbers. In 1987, Dan Gordon defined the notion of an elliptic Carmichael number as a composite integer n which satisfies a Fermat- like criterion on elliptic curves with complex multiplication. More recently, in 2018, Thomas Wright showed that there are infinitely such numbers. We build off the work of Wright to prove that there are infinitely many elliptic Carmichael numbers of the form a (mod M) for a certain M, using an improved lower bound due to Carl Pomerance. We then apply this result to comment on the infinitude of strong pseudoprimes and strong Lucas pseudoprimes.</p><p><br></p><p>Finally, we consider the problem of classifying for which k does one have Φk(x) | Φn(x)−1, where Φn(x) is the nth cyclotomic polynomial. We provide a motivating example as to how this can be applied to primality proving. Then, we complete the case k = 8 and give a partial characterization for the case k = 16. This leads us to conjecture necessary and sufficient conditions for when Φk(x) | Φn(x) − 1 whenever k is a power of 2.</p></div></div></div>