A new mathematical preprint gives an explicit asymptotic formula for a mixed second moment built from quadratic Dirichlet L-functions. The quantity combines the second derivative of the completed L-function at the central point with the L-function itself, then averages that product over monic irreducible polynomials in a function-field setting. The paper’s main result describes how this normalized average expands as the degree parameter tends to infinity.
The leading displayed term is g⁵/(15ζ_A(2)). Two further displayed orders follow: q-dependent corrections multiplied by g⁴ and g³, followed by a stated error term. The result is therefore more detailed than a single growth rate: it records the leading power and the next displayed corrections in the average.
A second result for the uncompleted function
The paper also states a corollary for the corresponding uncompleted mixed moment. Its leading term is 2g⁵/(5ζ_A(2)), followed by a g⁴ term with coefficient 2+7/(3q), and then a stated error term. The two formulas describe related but separately defined versions of the averaged quantity.
The authors frame Theorem 1.3 as the monic irreducible version of Theorem 1.1. That description highlights the defining feature of the averaging family in this result: the polynomials are required to be monic and irreducible.
A family that grows with the problem
The average runs over all monic irreducible, or “prime,” polynomials P of degree 2g+1 in Fq[t]. The main theorem assumes q is an odd prime and takes the limit g→∞. This is a changing polynomial family indexed by its degree parameter, rather than a result stated for one fixed finite collection.
The central object is not a single value of an L-function but a normalized family average involving the function and its second derivative at the central point. That focus makes the paper’s question one of describing the family’s behavior as g grows.
How the calculation is built
The proof starts with approximate functional equations for the polynomial-family L-functions. It uses the completed L-function’s symmetric functional equation and the fact that its first derivative vanishes at the central point, leaving the second derivative as the derivative term in the mixed moment.
To control the sums that appear, the argument applies a Weil bound to character sums over monic irreducible polynomials. It also invokes an upper bound for moments of Dirichlet L-functions. These estimates are the analytic tools used to handle the polynomial family and the L-function factors in the average.
The calculation then separates a main contribution S(X) from a tail contribution E(X). The tail analysis applies the function-field analogue of Perron’s formula and breaks E(X) into four contour contributions, E1(X), E2(X), E3(X) and E4(X). The tail calculation produces an explicit polynomial expansion in g and g−X, together with error terms.
A matching main-term calculation gives S(X) an explicit polynomial expression, with tail-shaped terms subtracted and additional error terms. The two parts provide the detailed expansions from which the displayed asymptotic formula is assembled.
A result with a defined boundary
The formula’s boundary is clear. It is an asymptotic result in the limit g→∞, and its main theorem is restricted to odd-prime q and monic irreducible polynomials of degree 2g+1. The statement therefore applies to the specified prime-polynomial family and parameter range.
For general readers, the key point is the level of resolution: the preprint does not merely identify a dominant g⁵ scale. It also gives explicit q-dependent terms at lower displayed powers for both the completed and uncompleted mixed moments.
The work remains a preprint. The supplied record identifies it as arXiv version 1, dated 26 August 2026.
Paper data and sources
Original title: The Mixed Second Moment of Quadratic Dirichlet $L$-functions with Prime Conductors
Authors: J. Macmillan
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text