Proof of the functional equation for the Riemann zeta-function

In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.

EULER PRODUCT ASYMPTOTICS FOR DIRICHLET -FUNCTIONS

The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.

TWISTED DOUBLING INTEGRALS FOR BRYLINSKI–DELIGNE EXTENSIONS OF CLASSICAL GROUPS

We explain how to develop the twisted doubling integrals for Brylinski–Deligne extensions of connected classical groups. This gives a family of global integrals which represent Euler products for this class of nonlinear extensions.

Low Pseudomoments of Euler Products

In this paper, we determine the order of magnitude of the 2 q-th pseudomoment of powers of the Riemann zeta function $\zeta(s)^{\alpha}$ for $0 \lt q\le 1/2$ and $0 \lt \alpha \lt 1$, completing the results of Bondarenko, Heap and Seip, and Gerspach. Our results also apply to more general Euler products satisfying certain conditions.

Harmonic analysis and statistics of the first Galois cohomology group

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $$f\in H^1(K,T)$$ f ∈ H 1 ( K , T ) with discriminant-like invariant $$\text {inv}(f)\le X$$ inv ( f ) ≤ X as $$X\rightarrow \infty $$ X → ∞ . Specifically, Poisson summation produces a canonical decomposition for the corresponding generating series as a sum of Euler products for a very general counting problem. This type of decomposition is exactly what is needed to compute asymptotic growth rates using a Tauberian theorem. These new techniques allow for the removal of certain obstructions to known results and answer some outstanding questions on the generalized version of Malle’s conjecture for the first Galois cohomology group.