Selberg's Formula Research Articles

Lambert series of logarithm, the derivative of Deninger’s function $R(z),$ and a mean value theorem for $\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )$

AbstractAn explicit transformation for the series $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ , or equivalently, $\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re $(y)>0$ , which takes y to $1/y$ , is obtained for the first time. This series transforms into a series containing the derivative of $R(z)$ , a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi _1(z)$ (the derivative of $R(z)$ ) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$ , all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as $y\to 0$ which was also not known before. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )e^{-\delta t}\, dt$ as $\delta \to 0$ .

Comparison of the empirical formula and the “CFD”-based semi-empirical method in the prediction of the critical speed of flutter

In this paper, the critical wind speed of flutter has been predicted by using two-dimensional computational fluid dynamics (CFD) model side by side with the empirical formula (Selberg's formula, 1961). The deck section of the Great Belt Bridge's main span has been used herein as a reference among other sectional varieties. For each case of the examined sections, the width was changed case by case, while its depth maintained unchanged. Both of the two methods, the semi-empirical using the “CFD” simulation and the empirical formula have been employed to predict the critical wind speed of flutter for a variety of the (width/depth) ratio and the natural frequency ratio (torsional/vertical). The results were obtained from the semi-empirical method and validated by comparison with wind tunnel test for typical section model of the Great Belt Bridge presented in the literature, and also compared with the empirical method. The results were represented graphically for each method and they have indicated the variation of the critical speed in terms of the section dimension and its natural frequencies, and furthermore, the change of the difference between the two methods was represented too.

Machine learning strategy for predicting flutter performance of streamlined box girders

Engineers often heavily rely on wind tunnel tests or computational fluid dynamics (CFD) to evaluate the flutter performance of bridges in their preliminary design, which is costly and time-consuming. To quickly obtain the critical flutter wind speed of streamlined box girders in the preliminary design, a machine learning (ML) strategy was proposed in this paper. A big dataset was built by testing critical flutter wind speeds of 30 sectional models of streamlined box girders with and without railings at 5 angles of attack through free vibration wind tunnel tests. The flutter predicting models, taking geometric information and dynamic parameters as inputs, were built based on four widely-used ML algorithms, i.e., support vector regression (SVR), neural network (NN), random forest regression (RFR), and gradient boosting regression tree (GBRT). It is shown that the NN and GBRT models exhibit the highest prediction accuracy for the girders regardless of railings, respectively. A comparative study revealed that the ML models were superior over those simplified formulas for flutter estimation including the Van der Put formula, Selberg formula, and Haifan Xiang formula. A case study was also given to demonstrate the practical application of the proposed method. These ML models provide an efficient supplement to wind tunnel tests and CFD simulations for flutter predictions of streamlined box girders in the preliminary design.

Remarks on the Number Theory Over Additive Arithmetical Semigroups

We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding ζ-functions. We use these results to prove prime number theorems and a Selberg formula for semigroups of this kind.

Remarks on number theory over additive arithmetical semigroups

UDC 511 We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the correspondingζ-functions. We use these results to prove prime number theorems and a Selberg formula for such semigroups.

The function field Sathe--Selberg formula in arithmetic progressions and `short intervals'

We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in $\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the limit as $n$ tends to infinity. We then adapt this method to count such polynomials in arithmetic progressions and short intervals, and by making use of Weil's `Riemann hypothesis' for curves over $\mathbb{F}_q$, obtain better ranges for these formulae than are currently known for their analogues in the number field setting. Finally, we briefly discuss the regime in which $q$ tends to infinity.

On the Colmez conjecture for non-abelian CM fields

The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of Artin L-functions at $$s=0$$ . In this paper, we prove that if F is any fixed totally real number field of degree $$[F:\mathbb {Q}] \ge 3$$ , then there are infinitely many effective, “positive density” sets of CM extensions E / F such that $$E/\mathbb {Q}$$ is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at arbitrary prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be viewed as an explicit non-abelian Chowla–Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang.

Hecke structures of weakly holomorphic modular forms and their algebraic properties

Let Sk!(Γ1(N)) be the space of weakly holomorphic cusp forms of weight k on Γ1(N) with an even integer k>2 and Mk!(Γ1(N)) be the space of weakly holomorphic modular forms of weight k on Γ1(N). Further, let z denote a complex variable and D:=12πi∂∂z. In this paper, we construct a basis of the space Sk!(Γ1(N))/Dk−1(M2−k!(Γ1(N))) consisting of Hecke eigenforms by using the Eichler–Shimura cohomology theory. Further, we study algebraicity of CM values of weakly holomorphic modular forms in the basis. This applies to an analogue of the Chowla–Selberg formula for a mock modular form whose shadow is the Ramanujan delta function.

Flutter characteristics of twin-box girder bridges with vertical central stabilizers

It has been well known that Vertical Central Stabilizers (VCS) have the potential of improving flutter performance of long-span bridges. However, the fundamental flutter mechanisms of VCS are still not fully understood so far. In this study, a series of wind-tunnel tests involving the combination of six representative heights and four types of VCS were conducted to fundamentally investigate the influence of VCS on flutter performance of twin-box girders with various Slot Width Ratios (SWRs). Experimental results show that the flutter instability of 20% SWR is significantly sensitive to the height change of VCS, whereas the VCS have little effect on the flutter performance for 80% and 100% SWR. In addition, the results from Two-Dimensional Three Degree of freedom (2D-3DOF) flutter analysis demonstrates that aerodynamic damping Part A with reference of flutter derivative A2∗ makes the greatest contribution to the flutter instability for a 0.8h/H VCS, while the role of Part D with reference of A1∗H3∗ becomes critical for a short VCS (i.e. the ratio of h/H is less than 0.2). Besides, the results of Computational Fluid Dynamics simulation indicate that the geometry of VCS could potentially influence the transforming vortices’ structures and pressure distribution under the central slotting. Finally, the modified Selberg formula presented in this study has the capability of predicting the critical flutter speeds of twin-box girders with various SWRs and VCS.

The Chowla–Selberg formula for abelian CM fields and Faltings heights

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

On Special Values of Jacobi-Sum Hecke L-Functions

For motives associated with Fermat curves, there are elements in motivic cohomology whose regulators are written in terms of special values of generalized hypergeometric functions. Using them, we verify the Beilinson conjecture numerically for some cases and find formulas for the values of L-functions at 0. These appear analogous to the Chowla–Selberg formula for the periods of elliptic curves with complex multiplication, which are related to the L-values at 1 by the Birch–Swinnerton-Dyer conjecture.

Aerodynamic Stabilization Mechanism of a Twin Box Girder with Various Slot Widths

Five representative girder cross sections with various slot widths are utilized to analyze the effects of center slots on their aerodynamic performance, based on wind-tunnel tests and theoretical analyses. It is shown that the favorable aerodynamic effects of the center slot on bridge decks depend on the aerodynamic shape of the box girders and on the slot widths rather than unconditionally improving the aeroelastic stability. Further investigation of a streamlined box girder with various slot widths results in a modified Selberg formula to calculate the critical flutter wind speed for design purposes, wherein the Lorentz peak-value function is utilized. The flutter mechanism is illustrated utilizing a two-dimensional three-degrees-of-freedom (2D-3DOF) analysis scheme. The results indicate that the center slot changes the participation level of the heaving motion at the flutter onset, which is highly correlated with the critical flutter wind speed. In addition, particle image velocimetry (PIV) and proper orthogonal decomposition (POD) techniques are employed to assist in revealing the aerodynamic stabilization mechanism of the center slotting of box girders.

On gamma quotients and infinite products

Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note is to demonstrate the usefulness of this fact through a number of diverse applications involving multiplicative partitions, entries in Ramanujanʼs notebooks, the Chowla–Selberg formula, and the Thue–Morse sequence. In addition, we propose a numerical method for efficiently evaluating more general infinite series such as the slowly convergent Kepler–Bouwkamp constant.

Flutter and its application—Flutter mode and ship navigation

Based on the aerodynamic interaction between the heaving vortex-induced vibration and the torsional flutter, another explanation of the drastic change of the vibrational mode from the 5th symmetrical heaving mode to the 1st asymmetrical torsional mode is introduced. Some fundamental characteristics of the coupled flutter of a thin plate and a plate like structure, such as the flutter modes, the flutter branches, are explained by use of the “Step-by-Step flutter analysis”. The flutter properties obtained by the 3DOF multi-modes-flutter analysis are compared with the ones by the 2DOF 2modes- flutter analysis t. 3DOF multi-modes flutter analysis is recommended to use for more accurate verification of the flutter property as the lessons from the aerodynamic behavior of the full-scale elastic model of Akashi Kaikyo Bridge. The reduction of the key equivalent aerodynamic derivatives, by suitable span-wise arrangement of the girders with different girder-shapes, is significantly effective to stabilize the flutter instability of a long spanned suspension bridge. Furthermore, the significantly similar formula with the Selberg formula can be derived under the simple assumptions. As an application of the coupled flutter, the generation mechanism of the propulsion force by use of coupled flutter motion is introduced. Finally, it is confirmed that the H-90 mode can produce the propulsion force by the test of the ship-navigation with the flapping plate controlled in the particular flutter mode is shown.

CASIMIR EFFECT WITH A HELIX TORUS BOUNDARY CONDITION

We use the generalized Chowla–Selberg formula to consider the Casimir effect of a scalar field with a helix torus boundary condition in the flat (D + 1)-dimensional spacetime. We obtain the exact results of the Casimir energy density and pressure for any D for both massless and massive scalar fields. The numerical calculation indicates that once the topology of spacetime is fixed, the ratio of the sizes of the helix will be a decisive factor. There is a critical value r c of the ratio r of the lengths at which the pressure vanishes. The pressure changes from negative to positive as the ratio r passes through r c increasingly. In the massive case, we find the pressure tends to the result of massless field when the mass approaches zero. Furthermore, there is another critical ratio of the lengths [Formula: see text] and the pressure is independent of the mass at [Formula: see text] in the D = 3 case.

The Chowla–Selberg Formula and The Colmez Conjecture

AbstractIn this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

New consideration on flutter properties based on step-by-step analysis

This paper studies the coupled flutter mechanism of plate and long span bridges based on step-by-step analysis (SBS). Fundamental flutter modes are defined based on amplitude ratio and phase difference between heaving and torsional motions. Furthermore, a formula remarkably similar to the Selberg formula can be derived by use of the particular simplified flutter-onset condition. In the process of SBS analysis, some torsional divergent velocities where the torsional rigidity becomes zero can be defined. Finally, the flutter-behavior of an elastic model of the complete Akashi-Kaikyo Bridge, which is the longest suspension bridge in the world, is studied from the point of view of flutter in 2 degrees of freedom, namely heaving and torsional motion, taking into account the structural coupling effect of additional torsional displacement induced by horizontal displacement as a structural coupling property.

Predicting flutter speed of a cable-stayed bridge

Cable-stayed bridges are prone to wind-induced instabilities owing to their smaller frequencies. Further, because of the closeness of torsional and vertical bending frequencies they are more likely to suffer from classical flutter. In this paper, empirical formulae have been proposed for estimating the basic frequencies as well as the flutter critical wind speed. Corroborative examples of a few cable-stayed bridges of different spans, namely, Karkinen, Nanpu, Yangpu and Oresund, have been considered. The flutter critical speeds of these bridges by the proposed method have been compared with those obtained from Selberg's formula and the Chinese method. Lastly, based on the safety margins with respect to the critical flutter speeds, cable-stayed bridges have been categorised into three groups, and the need for wind tunnel tests has been proposed. It is expected that the approach provided here will serve as a ready reckoner for cable-stayed bridge designers and will help reduce the project cost and time by assessing the flutter susceptibility of bridges.

On a generalization of the Selberg formula

In this paper, we prove a theorem related to the asymptotic formula for ψ k ( x ; q , a ) which is used to count numbers up to x with at most k distinct prime factors (or k-almost primes) in a given arithmetic progression a ( mod q ) . This theorem not only gives the asymptotic formula for ψ k ( x ; q , a ) (or Selberg formula), but has played an essential role, recently, in obtaining a lower bound for the variance of distribution of almost primes in arithmetic progressions.

Zeros of the constant term in the Chowla–Selberg formula

Zeros of the constant term in the Chowla–Selberg formula