Riemann Zeta Function: Integrals, Multiple Sums, Mordell's Sums, Ramanujan's Formula

The Riemann zeta function plays a central role in many areas in which complex analysis is applied, such as number theory (e.g. generating irrational and prime numbers. It is also an important tool in signal analysis in many fields of contemporary practice and technology, cryptography. In condensed matter physics, for example, the famous Sommerfeld expansion, which is used to calculate the number of particles and the internal electron energy, includes the Riemann zeta function with even integer argument values. On the other hand, the spin-spin correlation function of isotropic spin-1/2 in the Heisenberg model is expressed by ln 2 and Riemann zeta function with odd integer arguments. The author has made a tremendous effort to provide the reader with a new, clear and innovative way of looking at the most important features of the Riemann zeta function. The proofs of the expressed theorems are completely original. The monography established a good theoretical basis for the problem of calculating multiple sums and integrals in which the Riemann function appears. A special method was developed to establish a connection between the values of the Riemann zeta function with odd and even integer arguments. Based on the results obtained by H. M. Srivastava in his study from 1988, related to several different groups of summation formulas with the series in which the Riemann zeta function appears (which was first investigated by Euler and Goldbach), the monograph dealt with the series of this type. A new formula will be offered for Riemann function for odd argument which has a more compact form and faster convergence than any of the relations described in the afore mentioned papers.