f(n) exceeds (C-o(1)) log n for any fixed C>1 and infinitely many n, so limsup f(n)/log n is infinite.
New large value estimates for Dirichlet polynomials
6 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length $N$ taking values of size close to $N^{3/4}$, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate $N(\sigma,T)\le T^{30(1-\sigma)/13+o(1)}$ and asymptotics for primes in short intervals of length $x^{17/30+o(1)}$.
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Asymptotics are derived for the number of integers in bad or very bad consecutive intervals, with near-asymptotics for type F3 interval endpoints and solutions to a1! a2! a3! = m².
Under the Generalized Density Hypothesis, the prime number theorem holds in shorter intervals than the classic bounds for arithmetic progressions with moduli up to log powers of x.
Most short intervals of length X^theta (theta > 2/15 + eps) contain asymptotically h integers of the form p + a with p prime and a in the lacunary set A_lambda(X).
For irrationals α with bounded continued fraction terms, the number of primes p in (X-Y, X] with ||pα|| < δ is asymptotically 2δ Y / log X when X^{2/3+ε} ≤ Y ≤ X/2 and δ satisfies the given lower bound.
Claims to prove equivalence between Riemann zeta zero distributions and those of a 2D Ising model with mixed ferromagnetic and random competing interactions, concluding the zeros lie on the critical line.
citing papers explorer
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Unbounded logarithmic limsup in Erd\H{o}s problem 684
f(n) exceeds (C-o(1)) log n for any fixed C>1 and infinitely many n, so limsup f(n)/log n is infinite.
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Products of consecutive integers with unusual anatomy
Asymptotics are derived for the number of integers in bad or very bad consecutive intervals, with near-asymptotics for type F3 interval endpoints and solutions to a1! a2! a3! = m².
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Refinements for primes in short arithmetic progressions
Under the Generalized Density Hypothesis, the prime number theorem holds in shorter intervals than the classic bounds for arithmetic progressions with moduli up to log powers of x.
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Short intervals for the Romanoff-type sumset
Most short intervals of length X^theta (theta > 2/15 + eps) contain asymptotically h integers of the form p + a with p prime and a in the lacunary set A_lambda(X).
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Diophantine approximation with primes from short intervals
For irrationals α with bounded continued fraction terms, the number of primes p in (X-Y, X] with ||pα|| < δ is asymptotically 2δ Y / log X when X^{2/3+ε} ≤ Y ≤ X/2 and δ satisfies the given lower bound.
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Equivalence between the zero distributions of the Riemann zeta function and a two-dimensional Ising model with randomly distributed competing interactions
Claims to prove equivalence between Riemann zeta zero distributions and those of a 2D Ising model with mixed ferromagnetic and random competing interactions, concluding the zeros lie on the critical line.