Montgomery’s 100-percent critical-line conjecture for linear combinations of L-functions

Prove that a linear combination of primitive Dirichlet L-functions satisfying the same functional equation and the required independence conditions has 100% of its non-trivial zeros on the critical line.

Background

Linear combinations of primitive L-functions can lack an Euler product and may have zeros away from the critical line. The paper discusses Montgomery’s conjecture that, under natural compatibility conditions—especially a common functional equation and an appropriate independence condition—almost all, indeed 100%, of their non-trivial zeros lie on the critical line.

Bombieri and Hejhal established this conclusion conditionally under generalized Riemann hypotheses and a well-spacing hypothesis. The paper proves only a positive unconditional proportion, specifically at least 31% for a linear combination of two distinct primitive Dirichlet L-functions with the same conductor and parity, so the 100% assertion remains unresolved.

References

Although a linear combination of $L$-functions might have many zeros off the critical line, Montgomery conjectured that a linear combination of primitive $L$-functions satisfying certain natural conditions (notably the same functional equation and a certain notion of independence) has $100\%$ of its non-trivial zeros on the critical line.

— Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet $L$-functions  (2610.26841 - Lamzouri, 19 Oct 2026) in Section 1, subsection “Application to zeros of linear combinations of two Dirichlet L-functions”