## Pair Correlation and Distribution of Prime Numbers |

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### Contents

Equivalence of three formulas | 5 |

Beyond Pair Correlation | 28 |

Equivalence of Higher Even Moments | 37 |

2 other sections not shown

### Common terms and phrases

2TAy Assume Hypothesis HL Assume Riemann Hypothesis Cauchy's inequality Chapter Co-log2T constant may depend CóX deduce defined double error term Euler's constant Gaussian Unitary Ensemble Goldston and Montgomery h log Hardy–Littlewood Hence hfdx holds uniformly homogeneous polynomial i-H+ implicit constant ip(x klog left hand side Lemma II.9 Lemma IV.3 Lemma V.1 Lemma VI.3 Let f(x log log log2 log2T loga logh logx main term max{h(x Montgomery and Soundararajan nía Numerical evidence O(KA Pair Correlation Conjecture plk(N polynomial positive even integer positive integer positive real number precise asymptotic formula Prime k-tuple Conjecture prime number theorem Proof quantitative form range result[k Riemann zeta function short int ip Soundararajan 14 Stirling's formula Strong Pair Correlation temp2 tempi Theorem IV.5 Theorem VI.7 TlogT Twin Prime Conjecture uk(N w|Qs Weak Pair Correlation x"log y)dy