Among the topics are modular functions and Eisenstein series, the Riemann zeta function
, Euler's formulas and functional equations, functional equations, a linear space of solutions, and the multidimensional Poisson summation formula.
Among the poles of the meromorphic function Z(s) are the roots p of the Riemann zeta function
in the critical strip 0 < [sigma]< 1, which is clear from Eq.
The functional equation for the Riemann zeta function
is shown to be equivalent to the classical sampling theorem.
This text proposes an approach towards proving the truth of the conjecture that is based in large part on analogies between the Riemann zeta function
as expressed analytically by a functional equation and the dualities exhibited by string theories in theoretical physics.
9), to the generalized Riemann Zeta function
On page 283 of The Music of the Primes (Harper Collins, 2003), Marcus du Sautoy notes that 42 has a connection with the Riemann zeta function
In this we use the Riemann zeta function
There are an infinite number of these beings, and the Riemann Hypothesis says of them, "All the non-trivial zeros of the Riemann zeta function
have real part one half.
They cover the elementary methods, Bernoulli numbers, including the Riemann Zeta function
and the Euler-MacLaurin sum formula, modular forms and Hecke's theory of modular forms, representations of numbers as sums of squares, including the singular series and Liouville's methods and elliptical modular forms, arithmetic progression, and applications such as computing sums of two to four squares, resonant cavities and diamond cutting.
18] Titchmarsh Edward Charles, Heath-Brown David Rodney, The Theory of the Riemann Zeta function
, Oxford, 1986.
Keywords Riemann zeta function
, Smarandache-Riemann zeta sequence, positive integer.
Abstract In this paper, some recursion formulae of sums for the Riemann Zeta function
and Dirchlet, series are obtained through expanding several simple function on [-[pi],[pi]] or [0, 2[pi]] by using the Dirichlet theorem in Fourier series theory.