The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

AbstractHurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials

A note on Appell sequences, Mellin transforms and Fourier series

A large class of Appell polynomial sequences {p n (x)} n=0 8 are special values at the negative integers of an entire function F(s, x), given by the Mellin transform of the generating function for the sequence. For the Bernoulli and Apostol-Bernoulli polynomials, these are basically the Hurwitz zeta function and the Lerch transcendent. Each of these have well-known Fourier series which are proved in the literature using various techniques. Here we find the latter Fourier series by directly calculating the coefficients in a straightforward manner. We then show that, within the context of Appell sequences, these are the only cases for which the polynomials have uniformly convergent Fourier series. In the more general context of Sheffer sequences, we find that there are other polynomials with uniformly convergent Fourier series. Finally, applying the same ideas to the Fourier transform, considered as the continuous analog of the Fourier series, the Hermite polynomials play a role analogous to that of the Bernoulli polynomials

Mathematical source references

This list of references is intended to be a convenient reference source for those interested in the historical origin of common mathematical ideas, The topics mentioned are mostly those met in a degree course in mathematics. For each entry the list attempts to give an exact source reference with comments about priority. There are now available other historical reference sources for mathematics on the internet but with a different style of presentation.<br/

Prime number races

Sous l’hypothèse de Riemann généralisée et l’hypothèse d’indépendance linéaire, Rubinstein et Sarnak ont prouvé que les valeurs de x > 1 pour lesquelles nous avons plus de nombres premiers de la forme 4n + 3 que de nombres premiers de la forme 4n + 1 en dessous de x ont une densité logarithmique d’environ 99,59%. En général, l’étude de la différence #{p < x : p dans A} − #{p < x : p dans B} pour deux sous-ensembles de nombres premiers A et B s’appelle la course entre les nombres premiers de A et de B. Dans ce mémoire, nous cherchons ultimement à analyser d’un point de vue numérique et statistique la course entre les nombres premiers p tels que 2p + 1 est aussi premier (aussi appelés nombres premiers de Sophie Germain) et les nombres premiers p tels que 2p − 1 est aussi premier. Pour ce faire, nous présentons au préalable l’analyse de Rubinstein et Sarnak pour pouvoir repérer d’où vient le biais dans la course entre les nombres premiers 1 (mod 4) et les nombres premiers 3 (mod 4) et émettons une conjecture sur la distribution des nombres premiers de Sophie Germain.Under the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, Rubinstein and Sarnak proved that the values of x which have more prime numbers less than or equal to x of the form 4n + 3 than primes of the form 4n + 1 have a logarithmic density of approximately 99.59%. In general, the study of the difference #{p < x : p in A} − #{p < x : p in B} for two subsets of the primes A and B is called the prime number race between A and B. In this thesis, we will analyze the prime number race between the primes p such that 2p + 1 is also prime (these primes are called the Sophie Germain primes) and the primes p such that 2p − 1 is also prime. To understand this, we first present Rubinstein and Sarnak’s analysis to understand where the bias between primes that are 1 (mod 4) and the ones that are 3 (mod 4) comes from and give a conjecture on the distribution of Sophie Germain primes

Superirreducibility of Polynomials, Binomial Coefficient Asymptotics and Stories from my Classroom

In the first main section of this thesis, I investigate superirreducible polynomials over fields of positive characteristic and also over Q and Z. An n-superirreducible polynomial f(x) is an irreducible polynomial that remains irreducible under substitutions f(g(x)) for g of degree at most n. I find asymptotics for the number of 2-superirreducible polynomials over finite fields. Over the integers, I give examples of both families of superirreducible polynomials and families of irreducible polynomials which have an obstruction to superirreducibility. The writing and results on finite fields in this section have come from a collaboration with Jonathan Bober, Dan Fretwell, Gene Kopp and Trevor Wooley. The results over Z and Q are my own independent work. In the second section I determine the asymptotic growth of certain arithmetic functions A(n), B(n) and C(n), related to digit sum expansions. I consider these functions as sums over primes p up to n. I obtain unconditional results as well as results with better error terms conditional on the Riemann Hypothesis. The results over primes have come from collaboration with Jeff Lagarias. I also independently solved the analogous problem of summing over all positive integers b ≤ n. Finally in the third section, I discuss mathematical education via the lens of interviews and interactions. I consider my role as a teacher through multiple real-life anecdotes and what those stories have taught me. My interviews were conducted with young mathemati- cians from Bronx, NY that I got the opportunity to talk to as a result of my employment with Bridge to Enter Advanced Mathematics during the summer of 2019. The anecdotes I give are from working with teenaged students from a variety of different cultural, socio- economical and mathematical backgrounds.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162876/1/hjkl_1.pd