The constant [gamma](p) appears naturally as the Euler constant for a p-analogue of the Riemann zeta function
(Kurokawa-Wakayama  (2004)), with the slight difference that our [gamma](p) equals their
are the corresponding values of the Riemann zeta function
i) to generalise efficient congruencing to approximately translation invariant systems, and explore consequent applications to Diophantine problems such as Waring~s problem, restriction problems from discrete Fourier analysis, and bounds for the Riemann zeta function
within the critical strip;
Based on courses written by mathematics professors at the University of Wurzburg for the August 2012 International Summer School, four papers explain how modularity can be useful when studying the asymptotic behavior of arithmetically interesting functions, the representation of real numbers by their continued fraction expansion or their expansion in some integer base, and the connections between multiplicative Toeplitz matrices and the Riemann zeta function
The aim of this paper is to show that the zeros of the Riemann zeta function
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.
n] can be evaluated using the Riemann Zeta function
j] is the generalized Riemann zeta function
known also as Hurwitz zeta function.
Bradley, Searching symbolically for Apery-like formulae for values of the Riemann zeta function
, SIGSAM Bulletin of Algebraic and Symbolic Manipulation, Association of Computing Machinery, 30 (1996), no.
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.