Divisible formal weight enumerators and extremal polynomials not satisfying the Riemann hypothesis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

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On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽

10.3390/math9111254 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1254

Author(s):

Xue Han ◽

Xiaofei Yan ◽

Deyu Zhang

Keyword(s):

Asymptotic Formula ◽

Fourier Coefficient ◽

Cusp Form ◽

Riemann Hypothesis ◽

Modular Group ◽

Fourier Coefficients ◽

Mean Value ◽

Symmetric Square ◽

The Mean ◽

Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

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Caustics of Poncelet Polygons and Classical Extremal Polynomials

Regular and Chaotic Dynamics ◽

10.1134/s1560354719010015 ◽

2019 ◽

Vol 24 (1) ◽

pp. 1-35 ◽

Cited By ~ 4

Author(s):

Vladimir Dragović ◽

Milena Radnović

Keyword(s):

Extremal Polynomials ◽

Poncelet Polygons

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Variants on Andrica’s Conjecture with and without the Riemann Hypothesis

Mathematics ◽

10.3390/math6120289 ◽

2018 ◽

Vol 6 (12) ◽

pp. 289 ◽

Cited By ~ 2

Author(s):

Matt Visser

Keyword(s):

Riemann Hypothesis ◽

Distribution Of Primes

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀n≥1, is p n + 1 - p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 /ln p n + 1 - p n /ln p n < 11/25; (n ≥ 1). Then, by considering more general mth roots, again assuming the Riemann hypothesis, I show that p n + 1 m - p n m < 44/(25 e[m < 2]); (n ≥ 3; m > 2). In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln2 pn + 1 - ln2 pn < 9; ln3 pn + 1 - ln3 pn < 52; ln4 pn + 1 - ln4 pn < 991; (n ≥ 1). I shall also update the region on which Andrica’s conjecture is unconditionally verified.

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