Models of Computation, Riemann Hypothesis, and Classical Mathematics

AbstractHuman endeavours span a wide spectrum of activities which includes solving fascinating problems in the realms of engineering, arts, sciences, medical sciences, social sciences, economics and environment. To solve these problems, classical mathematics methods are insufficient. The real-world problems involve many uncertainties making them difficult to solve by classical means. The researchers world over have established new mathematical theories such as fuzzy set theory and rough set theory in order to model the uncertainties that appear in various fields mentioned above. In the recent days, soft set theory has been developed which offers a novel way of solving real world issues as the issue of setting the membership function does not arise. This comes handy in solving numerous problems and many advancements are being made now-a-days. Jun introduced hybrid structure utilizing the ideas of a fuzzy set and a soft set. It is to be noted that hybrid structures are a speculation of soft set and fuzzy set. In the present work, the notion of hybrid ideals of a near-ring is introduced. Significant work has been carried out to investigate a portion of their significant properties. These notions are characterized and their relations are established furthermore. For a hybrid left (resp., right) ideal, different left (resp., right) ideal structures of near-rings are constructed. Efforts have been undertaken to display the relations between the hybrid product and hybrid intersection. Finally, results based on homomorphic hybrid preimage of a hybrid left (resp., right) ideals are proved.

Download Full-text

THE CONJECTURE IMPLIES THE RIEMANN HYPOTHESIS

Mathematika ◽

10.1112/s0025579316000139 ◽

2016 ◽

Vol 63 (1) ◽

pp. 29-33 ◽

Cited By ~ 3

Author(s):

Sandro Bettin ◽

Steven M. Gonek

Keyword(s):

Riemann Hypothesis

Download Full-text

ADDITIVE BASES AND NIVEN NUMBERS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972721000186 ◽

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.

Download Full-text

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.

Download Full-text

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.

Download Full-text