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Let Re(s)﹦π(n)／n,(n﹦2), and Im(s)﹦((π(n)﹣π(2))㏒n)i,(n＞2)， Riemann hypothesis is proved.
σ﹢ti is irrelevant to the distributive law of primes.
Integral and series transformations via Ramanujan's identities and Salem's type equivalences to the Riemann hypothesis
Mellin transform Riemann zeta-function Kontorovich-Lebedev transform modified Bessel functions Lambert transform
2012/6/6
We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of differe...
On Nicolas criterion for the Riemann Hypothesis
Nicolas inequality Euler totient Dedekind function Riemann Hypothesis
2010/12/31
Nicolas criterion for the Riemann Hypothesis is based on an inequality that Euler totient function must satisfy at primorial numbers. A natural approach to derive this inequality would be to prove tha...
A physics pathway to the Riemann hypothesis
Riemann hypothesis zeta function,quantum mechanics quantum chaos
2010/12/31
We present a brief review of the spectral approach to the Riemann hypothesis,according to which the imaginary part of the non trivial zeros of the zeta function are the the eigenvalues of the Hamilton...
Riemann hypothesis and Quantum Mechanics
Bost-Connes model Riemann Hypothesis Primorial numbers
2010/12/31
In their 1995 paper, Jean-Benoˆıt Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function ζ(β),where β is an inverse temperatu...
Riemann hypothesis and Quantum Mechanics
Bost-Connes model Riemann Hypothesis Primorial numbers
2010/12/31
In their 1995 paper, Jean-Benoˆıt Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function ζ(β),where β is an inverse temperatu...
Riemann hypothesis and some new asymptotically multiplicative integrals which contain the remainder of the prime-counting function $\pi(x)$
Riemann hypothesis asymptotically multiplicative integrals
2010/11/1
A new parametric integral is obtained as a consequence of the Riemann hypothesis. An asymptotic multiplicability is the main property of this integral