On the distribution of zeros of linear combinations of Euler products

Zeros of the Epstein zeta function to the right of the critical line

“…When h(D) ≥ 2, it was conjectured by Montgomery that almost all complex zeros of E(s, Q) lie on the critical line Re(s) = 1/2. This conjecture was proved by Bombieri and Hejhal [1] conditionally on the Generalized Riemann Hypothesis and a weak version of a pair correlation conjecture.…”

The Saddle-Point Method and the Li Coefficients

“…A F(z)dz, it is convenient to set F(z) = e f (z) , where f (z) ≡ f n (z), involves some large parameter n. We chose a contour C through a saddle-point η such that f ′ (η) = 0. Next, we split the contour as C = C (0) ∪ C (1) , and the following conditions are to be verified. (i) On the contour C (1) the tails integral C (1) is negligible…”

“…Next, we split the contour as C = C (0) ∪ C (1) , and the following conditions are to be verified. (i) On the contour C (1) the tails integral C (1) is negligible…”

“…Furthermore, the length of the contour is O( n log n ), and we obtain |I ′ L1 | = O (1). Let s = σ + it be a point on one of the two horizontal segments.…”

“…with suitable coefficients b(p k ) satisfying b(p k ) = O(p kθ ) for some θ < 1 2 . It is expected that for every function in the Selberg class the analogue of the Riemann hypothesis holds, i.e, that all non trivial (non-real) zeros lie on the critical line ℜ(s) = 1 2 . The degree of F ∈ S is defined by…”