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.

<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>