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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.
Some heuristics on the gaps between consecutive primes
, 2011
"... We propose the formula for the number of pairs of consecutive primes pn, pn+1 < x separated by gap d = pn+1−pn expressed directly by the number of all primes < x, i.e. by pi(x). As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecut ..."
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We propose the formula for the number of pairs of consecutive primes pn, pn+1 < x separated by gap d = pn+1−pn expressed directly by the number of all primes < x, i.e. by pi(x). As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than x, for the generalized Brun’s constants and the first occurrence of a given gap d. Also the leading term log log(x) in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.
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.
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
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
"... 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 Ba ..."
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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.
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.
Smarandache’s Conjecture on Consecutive Primes
 INTERNATIONAL J.MATH. COMBIN. VOL.4(2014), 6991
, 2014
"... Let p and q two consecutive prime numbers, where q> p. Smarandache’s conjecture states that the nonlinear equation qx − px = 1 has solutions> 0.5 for any p and q consecutive prime numbers. This article describes the conditions that must be fulfilled for Smarandache’s conjecture to be true. ..."
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Let p and q two consecutive prime numbers, where q> p. Smarandache’s conjecture states that the nonlinear equation qx − px = 1 has solutions> 0.5 for any p and q consecutive prime numbers. This article describes the conditions that must be fulfilled for Smarandache’s conjecture to be true.
A New Sifting function J_n+1 (ω) in Prime Distribution
"... We define that prime equations 1 1 1 ( , ,), , ( ,)n k nf P P f P PL L L （5） are polynomials (with integer coefficients) irreducible over integers, where 1, , nP PL are all prime. If sifting function 1 ( ) 0nJ ω+ = then （5）has finite prime solutions. If 1 ( ) 0nJ ω+ ≠ then there are infin ..."
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We define that prime equations 1 1 1 ( , ,), , ( ,)n k nf P P f P PL L L （5） are polynomials (with integer coefficients) irreducible over integers, where 1, , nP PL are all prime. If sifting function 1 ( ) 0nJ ω+ = then （5）has finite prime solutions. If 1 ( ) 0nJ ω+ ≠ then there are infinitely many primes 1, , nP PL such that 1, kf fL are primes. We obtain a unite prime formula in prime distribution primes}are,,:,,{)1, ( 111 kffNPPnN knk LL ≤=++π