Tom M. Apostol

Another Elementary Proof of Euler's Formula for ζ(2n)

(1973). Another Elementary Proof of Eulers Formula for ζ(2n) The American Mathematical Monthly: Vol. 80, No. 4, pp. 425-431.

Dirichlet series related to the Riemann zeta function

Abstract A study is made of the function H(s, z) defined by analytic continuation of the Dirichlet series H(s, z) = Σn=1∞ n−s Σm=1n m−z, where s and z are complex variables. For each fixed z it is shown that H(s, z) exists in the entire s-plane as a meromorphic function of s, and its poles and residues are determined. Also, for each fixed s ≠ 1 it is shown that H(s, z) exists in the entire z-plane as a meromorphic function of z, and again its poles and residues are determined. Two different representations of H(s, z) are given from which a reciprocity law, H(s, z) + H(z, s) = ζ(s) ζ(z) + ζ(s + z), is deduced. For each integer q ≥ 0 the function values H(s, −q) and H(−q, s) are expressed in terms of the Riemann zeta function. Similar results are also obtained for the Dirichlet series T(s, z) = Σn=1∞ n−s Σm=1n m−z (m + n)−1. Applications include identities previously obtained by Ramanujan, Williams, and Rao and Sarma.

A proof that euler missed: evaluating ζ(2) the easy way

R. Apery [1] was the first to prove the irrationality of

A New Method for Investigating Euler Sums

\zeta \left( 3 \right) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^3}}}}

Irrationality of The Square Root of Two—A Geometric Proof

(4.1) .

Identities for sums of Dedekind type

Abstract A new representation for Dirichlet L -functions L ( s , χ ), valid for primitive characters χ modulo k and all complex s , is given in terms of the function F ( x , s ) defined for real x and R ( s ) > 1 by the series Σ n=1 ∞ e 2nπix n s . Evaluation of L ( s , χ ) for negative integer s leads to a class of identities relating m th power moments Σ r =1 k −1 χ ( r ) r m with finite cotangent power sums. Special emphasis is given to the quadratic character χ(n) = ( n p ) , p an odd prime. A new proof of the functional equation for L -functions is also given.

Unwrapping Curves from Cylinders and Cones

Euler discovered a recursion formula for the Riemann zeta function evaluated at the even integers. He also evaluated special Dirichlet series whose coefficients are the partial sums of the harmonic series. This paper introduces a new method for deducing Eulers formulas as well as a host of new relations, not only for the zeta function but for several allied functions.

Sums of Squares of Distances in m-Space

Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: