Extreme biases in prime number races with many contestants

Ford, Kevin; Harper, Adam J.; Lamzouri, Youness

doi:10.1007/s00208-019-01810-x

Cited by 6 publications

(4 citation statements)

References 13 publications

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“…The literature on prime number races which study such inequities is quite vast. See, for instance, the work of Fiorilli, Ford, Harper, Konyagin, Lamzouri and Martin [ FM13, FK02, FLK13, FHL19 ].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A zero density estimate and fractional imaginary parts of zeros for L-functions

Beckwith¹,

LIU²,

Thorner³

et al. 2022

Math. Proc. Camb. Phil. Soc.

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We prove an analogue of Selberg’s zero density estimate for $\zeta(s)$ that holds for any $\textrm{GL}_2$ L-function. We use this estimate to study the distribution of the vector of fractional parts of $\gamma\boldsymbol{\alpha}$ , where $\boldsymbol{\alpha}\in\mathbb{R}^n$ is fixed and $\gamma$ varies over the imaginary parts of the nontrivial zeros of a $\textrm{GL}_2$ L-function.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A zero density estimate and fractional imaginary parts of zeros for L-functions

Beckwith¹,

LIU²,

Thorner³

et al. 2022

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…However, note that the biases in Theorem 20 are always close to and smaller than 1, unlike the extreme biases predicted by Conjecture 16. In a recent preprint, Ford, Harper, and Lamzouri [16] improve on this by showing that the second part of the conjecture holds, actually as soon as n/ log q goes to ∞. More precisely,…”

Section: Orderings Of Weakly Correlated Normal Random Variables and Tmentioning

confidence: 95%

Some Recent Interactions of Probability and Number Theory

Perret-Gentil

2019

EMS Newsl.

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Around eighty years after its birth, the field of probabilistic number theory continues to see very interesting developments. On the occasion of a thematic program on the subject that took place last May in Montréal, we give a brief survey of a (far from exhaustive) selection of recent advances.for any z ∈ R. Along with the Erdős-Wintner theorem (1939) on limiting distributions of additive functions on the integers, this can be seen as the beginning of probabilistic number theory. We refer the reader to W. Schwarz's survey [44] for an account of the main developments that followed. Herein, we would like to give a brief survey of a (far from exhaustive) selection of recent works that use concepts such as martingales, suprema of Gaussian and log-correlated processes, orderings of weakly correlated random variables, normal approximations, large deviation estimates, comparison inequalities, and random Fourier series, to obtain significant results or insights in number theory. 1 A proof using the method of moments was given by Halberstam in 1955.

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“…In [3], Ford, Harper, and Lamzouri show that, although any ordering appears infinitely often, for n large with respect to q, the prime number races among orderings can exhibit large biases. They rely on the fact that counts of primes in distinct progressions have negative correlations, which they arrange to produce a bias.…”

Section: Toy Models and Open Problemsmentioning

confidence: 99%

“…We now want to evaluate the sum over γ 3 , a, and b (or in other words the sum over a 3 . We can split this sum into two cases: either ˇˇˇˇa3 which, as we saw in the proof of Lemma 2.6, gives T 3 ! h log h ´q φpqq ¯4, as desired.…”

mentioning

confidence: 99%

Odd moments in the distribution of primes

Kuperberg¹

2021

Preprint

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Montgomery and Soundararajan showed that the distribution of ψpx `Hq ψpxq, for 0 ď x ď N , is approximately normal with mean " H and variance " H logpN {Hq, when N δ ď H ď N 1´δ . Their work depends on showing that sums R k phq of k-term singular series are µ k p´h log h `Ahq k{2 `Ok ph k{2´1{p7kq`ε q, where A is a constant and µ k are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when k is odd, R k phq -h pk´1q{2 plog hq pk`1q{2 . We prove an upper bound with the correct power of h when k " 3, and prove analogous upper bounds in the function field setting when k " 3 and k " 5. We provide further evidence for this conjecture in the form of numerical computations.

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Extreme biases in prime number races with many contestants

Cited by 6 publications

References 13 publications

A zero density estimate and fractional imaginary parts of zeros for L-functions

A zero density estimate and fractional imaginary parts of zeros for L-functions

Some Recent Interactions of Probability and Number Theory

Odd moments in the distribution of primes

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