Period functions associated to real-analytic modular forms

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals of length one.

Performance of Different Risk Indicators in a Multi-Period Polynomial Portfolio Selection Problem Based on the Credibility Measure

In this paper, we study the portfolio selection problem considering transaction costs under multiple periods. For non-professional investors, it is a critical factor to choose an appropriate model among multiple portfolio selection models in investment. Based on the credibility measure, we formulate a multi-period polynomial portfolio selection model to gather the risk indicators involving variance, semi-variance, entropy, and semi-entropy, helping investors bet on assets. According to the polynomial goal programming (PGP) approach, investors can conquer the fields by combining apposite indicators to build appropriate models. Subsequently, an adjusted genetic algorithm on the foundation of the penalty function is designed to obtain the optimal solution of this multi-period model. The results indicate that the PGP method is suitable for investors to choose the model and assigns the proper models to investors with different risk preferences.

Period polynomial relations of binomial coefficients and binomial realization of formal double zeta space

In this paper, we give a realization of the formal double zeta space by using binomial coefficients. Along with the results in [Period polynomial relations between formal double zeta values of odd weight, Math. Ann. 365 (2016) 345–362], this gives us two families of period polynomial relations among binomial coefficients. We also give another family of period polynomial relations among binomial coefficients which cannot be obtained from our binomial realization. At the end, some higher depth observation is provided.

Riemann hypothesis for period polynomials of modular forms

The period polynomial rf(z) for an even weight k≥4 newform f∈Sk(Γ0(N)) is the generating function for the critical values of L(f,s). It has a functional equation relating rf(z) to rf(−1Nz). We prove the Riemann hypothesis for these polynomials: that the zeros of rf(z) lie on the circle |z|=1/N. We prove that these zeros are equidistributed when either k or N is large.