scholarly journals The Lindelöf hypothesis for primes is equivalent to the Riemann hypothesis

2020 ◽  
Vol 148 (7) ◽  
pp. 2863-2875
Author(s):  
Steven M. Gonek ◽  
Sidney W. Graham ◽  
Yoonbok Lee
Keyword(s):  
Riemann Hypothesis ◽  
Lindelöf Hypothesis

scholarly journals On the variance of squarefree integers in short intervals and arithmetic progressions

2021 ◽  
Author(s):  
Ofir Gorodetsky ◽  
Kaisa Matomäki ◽  
Maksym Radziwiłł ◽  
Brad Rodgers
Keyword(s):  
Riemann Hypothesis ◽  
Full Range ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Squarefree Integers ◽  
Lindelöf Hypothesis

AbstractWe evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$ H < x 6 / 11 - ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$ q > x 5 / 11 + ε . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H < x^{2/3 - \varepsilon }$$ H < x 2 / 3 - ε and $$q > x^{1/3 + \varepsilon }$$ q > x 1 / 3 + ε . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ H ε in the full range $$H < x^{1 - \varepsilon }$$ H < x 1 - ε is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

Highly composite numbers and the Riemann hypothesis

2021 ◽  
Author(s):  
Jean-Louis Nicolas
Keyword(s):  
Riemann Hypothesis

scholarly journals THE CONJECTURE IMPLIES THE RIEMANN HYPOTHESIS

Mathematika ◽  
2016 ◽  
Vol 63 (1) ◽  
pp. 29-33 ◽  
Author(s):  
Sandro Bettin ◽  
Steven M. Gonek
Keyword(s):  
Riemann Hypothesis

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

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.

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

Mathematics ◽  
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.

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

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 289 ◽  
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.

scholarly journals Riemann Hypothesis and strongly Ramanujan complexes from GL

2016 ◽  
Vol 161 ◽  
pp. 281-297 ◽  
Author(s):  
Ming-Hsuan Kang
Keyword(s):  
Riemann Hypothesis

scholarly journals Bounding |ζ(½+it )| on the Riemann hypothesis

2010 ◽  
Vol 43 (2) ◽  
pp. 243-250 ◽  
Author(s):  
Vorrapan Chandee ◽  
K. Soundararajan
Keyword(s):  
Riemann Hypothesis

scholarly journals On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis

2012 ◽  
Vol 64 (2) ◽  
pp. 247-261 ◽  
Author(s):  
S. K. Sekatskii ◽  
S. Beltraminelli ◽  
D. Merlini
Keyword(s):  
Riemann Hypothesis ◽  
Riemann Ζ Function
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