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17
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 Mill’s Constant and a Note on Honaker’s Problem
 Journal of Integer Sequences
, 2005
"... In 1947 Mills proved that there exists a constant A such that ⌊A3n ⌋ is a prime for every positive integer n. Determining A requires determining an effective Hoheisel type result on the primes in short intervals—though most books ignore this difficulty. Under the Riemann Hypothesis, we show that the ..."
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In 1947 Mills proved that there exists a constant A such that ⌊A3n ⌋ is a prime for every positive integer n. Determining A requires determining an effective Hoheisel type result on the primes in short intervals—though most books ignore this difficulty. Under the Riemann Hypothesis, we show that there exists at least one prime between every pair of consecutive cubes and determine (given RH) that the least possible value of Mills’ constant A does begin with 1.3063778838. We calculate this value to 6850 decimal places by determining the associated primes to over 6000 digits and probable primes (PRPs) to over 60000 digits. We also apply the CramérGranville Conjecture to Honaker’s problem in a related context.
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.
DETERMINISTIC METHODS TO FIND PRIMES
"... Abstract. Given a large positive integer N, how quickly can one construct a prime number larger than N (or between N and 2N)? Using probabilistic methods, one can obtain a prime number in time at most log O(1) N with high probability by selecting numbers between N and 2N at random and testing each o ..."
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Abstract. Given a large positive integer N, how quickly can one construct a prime number larger than N (or between N and 2N)? Using probabilistic methods, one can obtain a prime number in time at most log O(1) N with high probability by selecting numbers between N and 2N at random and testing each one in turn for primality until a prime is discovered. However, if one seeks a deterministic method, then the problem is much more difficult, unless one assumes some unproven conjectures in number theory; brute force methods give a O(N 1+o(1) ) algorithm, and the best unconditional algorithm, due to Odlyzko, has a run time of O(N 1/2+o(1)). In this paper we discuss an approach that may improve upon the O(N 1/2+o(1)) bound, by suggesting a strategy to determine in time O(N 1/2−c) for some c> 0 whether a given interval in [N, 2N] contains a prime. While this strategy has not been fully implemented, it can be used to establish partial results, such as being able to determine the parity of the number of primes in a given interval in [N, 2N] in time O(N 1/2−c). 1.
Different Approaches to the Distribution of Primes
 MILAN JOURNAL OF MATHEMATICS
, 2009
"... In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction. ..."
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In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction.
Deterministic Percolation
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1999
"... This paper examines percolation questions in a deterministic setting. In particular, I consider R, the set of elements of Z² with greatest common divisor equal to 1, where two sites are connected if they are at distance 1. The main result of the paper proves that the infinite component has an asympt ..."
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This paper examines percolation questions in a deterministic setting. In particular, I consider R, the set of elements of Z² with greatest common divisor equal to 1, where two sites are connected if they are at distance 1. The main result of the paper proves that the infinite component has an asymptotic density. An “almost everywhere” sieve of J. Friedlander is used to obtain the result.
1 On Goldbach’s Conjecture
, 2002
"... It is shown that if every odd integer n> 5 is the sum of three primes, then every even integer n> 2 is the sum of two primes. A conditional proof of Goldbach’s conjecture, based on Cramér’s conjecture, is presented. Theoretical and experimental results available on Goldbach’s conjecture allow that a ..."
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It is shown that if every odd integer n> 5 is the sum of three primes, then every even integer n> 2 is the sum of two primes. A conditional proof of Goldbach’s conjecture, based on Cramér’s conjecture, is presented. Theoretical and experimental results available on Goldbach’s conjecture allow that a less restrictive conjecture than Cramér’s conjecture be used in the conditional proof. A basic result of the Maier’s paper on Cramér’s model is criticized. 1
Query Access Assurance in Outsourced Databases
 IEEE TRANSACTIONS ON SERVICES COMPUTING
"... Query execution assurance is an important concept in defeating lazy servers in the database as a service model. We show that extending query execution assurance to outsourced databases with multiple data owners is highly inefficient. To cope with lazy servers in the distributed setting, we propose q ..."
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Query execution assurance is an important concept in defeating lazy servers in the database as a service model. We show that extending query execution assurance to outsourced databases with multiple data owners is highly inefficient. To cope with lazy servers in the distributed setting, we propose query access assurance (QAA) that focuses on IObound queries. The goal in QAA is to enable clients to verify that the server has honestly accessed all records that are necessary to compute the correct query answer, thus eliminating the incentives for the server to be lazy if the query cost is dominated by the IO cost in accessing these records. We formalize this concept for distributed databases, and present two efficient schemes that achieve QAA with high success probabilities. The first scheme is simple to implement and deploy, but may incur excessive server to client communication cost and verification cost at the client side, when the query selectivity or the database size increases. The second scheme is more involved, but successfully addresses the limitation of the first scheme. Our design employs a few number theory techniques. Extensive experiments demonstrate the efficiency, effectiveness and usefulness of our schemes.