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An orthogonal test of the LFunctions Ratios Conjecture
"... ABSTRACT. We test the predictions of the Lfunctions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N → ∞ through the primes or N = 1 and k → ∞. We study the main and lower order terms in the 1level density. We provide evidence for the Ratios ..."
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ABSTRACT. We test the predictions of the Lfunctions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N → ∞ through the primes or N = 1 and k → ∞. We study the main and lower order terms in the 1level density. We provide evidence for the Ratios Conjecture by computing and confirming its predictions up to a power savings in the family’s cardinality, at least for test functions whose Fourier transforms are supported in (−2, 2). We do this both for the weighted and unweighted 1level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1level densities differ by a term of size 1 / log(k 2 N). Finally, we show that there is another way of extending the sums arising in the Ratios Conjecture, leading to a different answer (although the answer is such a lower order term that it is hopeless to observe which is correct). 1.
Determining Mills’ Constants and a note on Honaker’s problem
 8 (2005), Journal of Integer Sequences
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ORTHOGONALITY AND THE MAXIMUM OF LITTLEWOOD COSINE POLYNOMIALS
"... Abstract. We prove that if p = 2q +1 is a prime, then the maximum of a Littlewood cosine polynomial qX Tq(t) = aj cos(jt), aj ∈ {−1, 1}, j=0 on the real line is at least c1 exp(c2(log q) 1/2), with an absolute constant c1 and c2 = p (log 2)/8. In the last section we observe that the maximum modulus ..."
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Abstract. We prove that if p = 2q +1 is a prime, then the maximum of a Littlewood cosine polynomial qX Tq(t) = aj cos(jt), aj ∈ {−1, 1}, j=0 on the real line is at least c1 exp(c2(log q) 1/2), with an absolute constant c1 and c2 = p (log 2)/8. In the last section we observe that the maximum modulus of a Barker polynomial p of degree n on the unit circle of the complex plane is always at least √ n + p 1/3.
Prime numbers
"... · · · prime numbers “grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. ” Don Zagier Abstract. In my talk I will pose several questions about prime numbers. We will see that on the one hand some of the ..."
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· · · prime numbers “grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. ” Don Zagier Abstract. In my talk I will pose several questions about prime numbers. We will see that on the one hand some of them allow an answer with a proof of just a few lines, on the other hand, some of them lead to deep questions and conjectures not yet understood. This seems to represent a general pattern in mathematics: your curiosity leads to a study of “easy ” questions related with quite deep structures. I will give examples, suggestions and references for further study. This elementary talk was meant for freshman students; it is not an introduction to number theory, but it can be considered as an introduction: “what is mathematics about, and how can you enjoy the fascination of questions and insights?”
A note on gaps
, 2009
"... Let pk denote the kth prime and d(pk) = pk−pk−1, the difference between consecutive primes. We denote by Nǫ(x) the number of primes ≤ x which satisfy the inequality d(pk) ≤ (log pk) 2+ǫ, where ǫ> 0 is arbitrary and fixed, and by π(x) the number of primes less than or equal to x. In this paper, ..."
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Let pk denote the kth prime and d(pk) = pk−pk−1, the difference between consecutive primes. We denote by Nǫ(x) the number of primes ≤ x which satisfy the inequality d(pk) ≤ (log pk) 2+ǫ, where ǫ> 0 is arbitrary and fixed, and by π(x) the number of primes less than or equal to x. In this paper, we first prove a theorem that limx→ ∞ Nǫ(x)/π(x) = 1. A corollary to the proof of the theorem concerning gaps between consecutive squarefree numbers is stated.
which satisfy a
, 2009
"... the density of prime differences less than a given magnitude ..."
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Riemann Zeta function:
"... Abstract. A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in Re(s)> 1/2. Some identities in the ring of formal power ..."
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Abstract. A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in Re(s)> 1/2. Some identities in the ring of formal power series involving rook theory and continued fractions are developed.
Maximal Gaps Between Prime kTuples: A Statistical Approach
"... Combining the HardyLittlewood ktuple conjecture with a heuristic application of extremevalue statistics, we propose a family of estimator formulas for predicting maximalgapsbetweenprimektuples. Extensivecomputationsshowthattheestimator alog(x/a)−ba satisfactorily predicts the maximal gaps below ..."
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Combining the HardyLittlewood ktuple conjecture with a heuristic application of extremevalue statistics, we propose a family of estimator formulas for predicting maximalgapsbetweenprimektuples. Extensivecomputationsshowthattheestimator alog(x/a)−ba satisfactorily predicts the maximal gaps below x, in most cases within an error of ±2a, where a = Cklog k x is the expected average gap between the same type of ktuples. Heuristics suggest that maximal gaps between prime ktuples near x are asymptotically equal to alog(x/a), and thus have the order O(log k+1 x). The distributionofmaximalgapsaroundthe“trend”curvealog(x/a)isclosetotheGumbel distribution. We explore two implications of this model of gaps: record gaps between primes and Legendretype conjectures for prime ktuples. 1