The Chebotarev Density Theorem in Short Intervals and Some Questions of Serre

Balog, Antal; Ono, Ken

doi:10.1006/jnth.2001.2694

Cited by 38 publications

(60 citation statements)

References 13 publications

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“…For example, in the case of f (X) = X +a, we immediately conclude from the Erdős-Selfridge result [6] that with some constant c(a) ∈ (0, 1), see for example [1]; when a = 1, we can take any c(1) > 7/12).…”

Section: Commmentsmentioning

confidence: 86%

On squares in polynomial products

et al. 2008

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Section: Commmentsmentioning

confidence: 86%

On squares in polynomial products

et al. 2008

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“…The inequality (3.7) clearly cannot hold for N large enough if we choose η < δ(1−δ)/2, since n < (1 + η)N and, by a result of Balog and Ono [5] or of Alkan [1], the gapñ − n is O(n c ), for some c < 1. In conclusion, for N sufficiently large, the set S E,r,δ,η,N is empty.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Dirichlet $L$-functions, elliptic curves, hypergeometric functions, and rational approximation with partial sums of power series

Berndt

Kim

Zaharescu

2013

Mathematical Research Letters

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Abstract. We consider the Diophantine approximation of exponential generating functions at rational arguments by their partial sums and by convergents of their (simple) continued fractions. We establish quantitative results showing that these two sets of approximations coincide very seldom. Moreover, we offer many conjectures about the frequency of their coalescence. In particular, we consider exponential generating functions with real Dirichlet characters and with coefficients of L-functions of elliptic curves, where calculational data provide striking examples showing agreement for certain convergents of high index and gargantuan heights. Finally, we similarly examine hypergeometric functions; note that e is a special case of the latter.

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“…In this connection, more recently Balog and Ono [4] obtained striking nonvanishing results for the Fourier coefficients of newforms without complex multiplication regarding their short interval distribution (see also [1]). In the case of the gap function the first author proved that i f (n) f,φ φ(n) for almost all n where φ is essentially any function monotonically tending to infinity.…”

Section: Introductionmentioning

confidence: 99%