Dirichlet Polynomials and Entropy

A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+⋯+a22y+a11y+a00y for some n,a0,…,an∈N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir→Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R⩾0×R⩾0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.

Motohashi’s fourth moment identity for non-archimedean test functions and applications

Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic $L$-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet $L$-functions modulo $q$ weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length $q^{1/4}$. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic $L$-functions, which we also use to improve the best known subconvexity bounds for automorphic $L$-functions in the level aspect.

More than five-twelfths of the zeros of $$\zeta $$ are on the critical line

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $$(\mu \star \Lambda _1^{\star k_1} \star \Lambda _2^{\star k_2} \star \cdots \star \Lambda _d^{\star k_d})$$(μ⋆Λ1⋆k1⋆Λ2⋆k2⋆⋯⋆Λd⋆kd) is computed unconditionally by means of the autocorrelation of ratios of $$\zeta $$ζ techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al. (Commun Number Theory Phys 2:593–636, 2008) as well as Conrey and Snaith (Proc Lond Math Soc 3(94):594–646, 2007). This in turn allows us to describe the combinatorial process behind the mollification of $$\begin{aligned} \zeta (s) + \lambda _1 \frac{\zeta '(s)}{\log T} + \lambda _2 \frac{\zeta ''(s)}{\log ^2 T} + \cdots + \lambda _d \frac{\zeta ^{(d)}(s)}{\log ^d T}, \end{aligned}$$ζ(s)+λ1ζ′(s)logT+λ2ζ′′(s)log2T+⋯+λdζ(d)(s)logdT,where $$\zeta ^{(k)}$$ζ(k) stands for the kth derivative of the Riemann zeta-function and $$\{\lambda _k\}_{k=1}^d$${λk}k=1d are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (Res Number Theory 4:9, 2018), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

Large prime factors on short intervals

We show that for all large enough x the interval [x, x + x1/2 log1.39x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman’s sieve method.

THE TWISTED SECOND MOMENT OF THE DEDEKIND ZETA FUNCTION OF A QUADRATIC FIELD

We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length T1/11-∊. Our result generalizes a formula of Hughes and Young concerning the fourth moment of the Riemann zeta function.