Mathematics

Zeckendorf's Theorem states that every positive integer can be uniquely represented as a sum of non-adjacent Fibonacci numbers, indexed from 1,2,3,5,… . This has been generalized by many authors, in particular to constant coefficient fixed depth linear recurrences with positive (or in some cases non-negative) coefficients. In this work we extend this result to a recurrence with non-constant coefficients, a n+1 =n a n + a n−1 . The decomposition law becomes every m has a unique decomposition as ∑ s i a i with s i ≤i , where if s i =i then s i−1 =0 . Similar to Zeckendorf's original proof, we use the greedy algorithm. We show that almost all the gaps between summands, as n approaches infinity, are of length zero, and give a heuristic that the distribution of the number of summands tends to a Gaussian. Furthermore, we build a game based upon this recurrence relation, generalizing a game on the Fibonacci numbers. Given a fixed integer n and an initial decomposition of n=n a 1 , the players alternate by using moves related to the recurrence relation, and whoever moves last wins. We show that the game is finite and ends at the unique decomposition of n , and that either player can win in a two-player game. We find the strategy to attain the shortest game possible, and the length of this shortest game. Then we show that in this generalized game when there are more than three players, no player has the winning strategy. Lastly, we demonstrate how one player in the two-player game can force the game to progress to their advantage.

Ramanujan provided several results involving the modified Bessel function K z (x) in his Lost Notebook. One of them is the famous Ramanujan-Guinand formula, equivalent to the functional equation of the non-holomorphic Eiesenstien series on S L 2 (z) . Recently, this formula was generalized by Dixit, Kesarwani, and Moll. In this article, we first obtain a generalization of a theorem of Watson and, as an application of it, give a new proof of the result of Dixit, Kesarwani, and Moll. Watson's theorem is also generalized in a different direction using μ K z (x,λ) which is itself a generalization of K z (x) . Analytic continuation of all these results are also given.

Let K be a local field with residue characteristic p and let L/K be a totally ramified extension of degree p k . In this paper we show that if L/K has only two distinct indices of inseparability then there exists a uniformizer ? L for L whose minimum polynomial over K has at most three terms. This leads to an explicit classification of extensions with two indices of inseparability. Our classification extends work of Amano, who considered the case k=1 .

We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r=1,2,3,4 , and partly classifies them, where the classification is complete for r=2,3,4 ; in fact, we show that there exists no normalized extremal quasimodular forms of depth 4 with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.

In this paper, we give an approximate formula for the measure of extreme values for the logarithm of the Riemann zeta-function and its iterated integrals. The result recovers the unconditional best result for the Ω -result of S 1 (t) for the part of minus of Tsang.

Let S(t)= 1 ? Imlogζ( 1 2 +it) . Using Soundararajan's resonance method we prove an unconditional lower bound on the size of the tails of the distribution of S(t) . In particular we reproduce the best unconditional Ω result for S(t) which is due to Tsang, S(t)= Ω ± ( ( logt loglogt ) 1/3 ), and get a bound on how often large values of S(t) occur. We also give a probabilistic argument for why this Ω result may be the best possible given our current knowledge of the zeros of the zeta function.

We provide a necessary and sufficient condition for n! to be a sum of three squares. The condition is based on the binary representation of n and can be expressed by the operation of an automaton.

Recently, Ni and Pan proved a q -congruence on certain sums involving central q -binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this q -congruence and confirm another q -congruence, also conjectured by Guo. Our proof uses Ni and Pan's technique and a simple q -congruence observed by Guo and Schlosser.

Let R be a subring of C[[z]] , and let X?�C[[z]] . The Newton-Puiseux Theorem implies that if the coefficients of X grow sufficiently rapidly relative to the coefficients of the series in R , then X is transcendental over R . We prove an alternative proof of this result by establishing a relationship between the coefficients of A(X) and A ??(X) , where A(T) is a polynomial over C[[z]] .

In this paper we study the Fermat equation x n + y n = z n over quadratic fields Q( d ??????) for squarefree d with 26?�d??7 . By studying quadratic points on the modular curves X 0 (N) , d -regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree d in this range there are no non-trivial solutions to this equation for n?? .