On Bernoulli identities and applications

In this paper, by using q-deformed bosonic p-adic integral, we give <TEX>$\lambda$</TEX>-Bernoulli numbers and polynomials, we prove Witt's type formula of <TEX>$\lambda$</TEX>-Bernoulli polynomials and Gauss multiplicative formula for <TEX>$\lambda$</TEX>-Bernoulli polynomials. By using derivative operator to the generating functions of <TEX>$\lambda$</TEX>-Bernoulli polynomials and generalized <TEX>$\lambda$</TEX>-Bernoulli numbers, we give Hurwitz type <TEX>$\lambda$</TEX>-zeta functions and Dirichlet's type <TEX>$\lambda$</TEX>-L-functions; which are interpolated <TEX>$\lambda$</TEX>-Bernoulli polynomials and generalized <TEX>$\lambda$</TEX>-Bernoulli numbers, respectively. We give generating function of <TEX>$\lambda$</TEX>-Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and <TEX>$\lambda$</TEX>-Bernoulli polynomials and ordinary Bernoulli numbers of order r and <TEX>$\lambda$</TEX>-Bernoulli numbers, respectively. We also study on <TEX>$\lambda$</TEX>-Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define <TEX>$\lambda$</TEX>-partial zeta function and interpolation function.

Five basic mistakes are found in the Riemann’s original paper proposed in 1859. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane and in the domain of the function, when the left-hand side of equation is finite, the right-hand side may be infinite, and vice versa. The Riemann Zeta function equation holds only at the point Re(s) = 1/2(s= a+ \(i b\)) . However, at this point, the Zeta function is infinite, rather than zero, to make Riemann hypothesis untenable. 2. An integral item around the original point of coordinate system was ignored when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) &gt; 1but divergent when Re(s) &lt; 1 .The integral form of Zeta function does not change the divergence of its series form. 3. A summation formula was used in the deduction of the integral form of Zeta function. The applicative condition of this formula is x &gt; 0 . At point x = 0 , the formula is meaningless. However, the lower limit of Zeta function’s integral is x = 0 , so the formula can not be used. 4. Because the integral lower limit of Zeta function is zero, the integrand function is not uniformly convergent, so integral sign and sum sign can not be exchanged. But Riemann made them interchangeable, resulting in that the integral form of Zeta function is untenable. 5.The formula \(\theta(x)=\sqrt{x} \theta(1 / x)\) of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula is also x &gt; 0 . Because the lower limit of integral in the deduction was x = 0 , this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, the analytic property of the original function is destroyed and the Cauchy-Riemann equation can not be satisfied. So they are not the real zeros of strict Riemann Zeta function.

A note on [formula omitted]-Bernoulli numbers and polynomials

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

Probability concepts and results are closely related to the study of zeros of the classical Riemann zeta function and its affinity to Gaussian and Gamma distributions. This is elaborated in obtaining the functional and integral equations for the zeta and in the determination first of the nonzero sets and then sets containing almost all (i.e., for the CLT probability measure) nontrivial zeros of the zeta function ζ(·). Also probability distributions determined by the zeta, based on the behavior of their finite dimensional distributions of the ζ(σ +it), σ > 0; as t varies and particularly the results of Denjoy, slightly sharpened, and also one of Salem are included. Several related opinions and comments are discussed.

In this paper, we give certain analytic functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell–Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950's.

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are uniformly distributed modulo p p for every p p . This is the basis of a well-known heuristic, given by Siegel, estimating the frequency of irregular primes. So far, analyses have shown that if Q ( D ) \mathbf {Q}(\sqrt {D}) is a real quadratic field, then the values of the zeta function ζ D ( 1 − 2 m ) = ζ Q ( D ) ( 1 − 2 m ) \zeta _{D}(1-2m)=\zeta _{\mathbf {Q}(\sqrt {D})}(1-2m) at negative odd integers are also distributed as expected modulo p p for any p p . We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.

On the Northcott property of zeta functions over function fields

Let ~ = G/K be a Hermitian symmetric space of the non-compact type, and let F C G be an arithmetic subgroup. It is obvious that the enveloping algebra U(g 0 of the complexified Lie algebra of G gives rise to operators which transform automorphic forms on N with respect to F to F-automorphic forms of a different type. However, it is in general not easy to write these differential operators explicitly in the coordinates of @. In the case G = Sp(n, IR), when N is the Siegel upper half plane of genus n, Maass constructed certain explicit operators of this type. As a consequence of his construction, he was able to show that his operators transform certain simple Eisenstein series for F to other specific simple Eisenstein series for F. For a summary of his results, and for references, see [6]. In this paper we generalize these results of Maass to arbitrary Hermitian symmetric spaces of the non-compact type. Namely, we show that the enveloping algebra U(g 0 gives rise to differential operators A such that, for certain pairs of canonical automorphy factors J, J ' on G, we have AJ=J': [cf. Theorem (2.4) and Corollary (3.4)]. We do this abstractly, using the transformation properties of the automorphy factors. In particular, we do not give explicit formulas for our differential operators, which we call Maass operators ; it would be interesting to do so in general. We have also restricted our attention to cases in which one of the automorphy factors J, J' is scalar-valued; the same methods probably work more generally. Our interest in these operators arose from their use by Shimura in his study of rationality properties for special values of zeta functions; see [8, 9], and subsequent papers. Katz [4] has also introduced these operators, in another form. The author has extended Shimura's method to the case of Siegel modular forms, in [3], and intends to use the results of the present paper in future work of the same kind. For this reason, a brief and somewhat obtuse discussion of the "rationality" of Maass operators has been included in Sect. 4. In any specific (i.e., coordinatized) example, the results of Sect. 4 can be improved upon.

Given a totally real fieldL of degreeg, we constructg Hasse invariants on Hilbert modular varieties in characteristicp and characterize their divisors. We show that these divisors give the type stratification defined by the action of\(\mathcal{O}_L \) on theαp-elementary subgroup. Under certain conditions, involving special values of zeta functions, the product of these Hasse invariants is the reduction of an Eisenstein series of weightp−1

This thesis deals with two properties of arithmetic functions, namely, factorization and independence. In 1984, Rearick pointed out that the ring of arithmetic functions is a unique factorization domain but is not a principal ideal domain and so is not a Euclidean domain. Without the Euclidean algorithm, the problem of factorizing arithmetic functions becomes quite difficult. In the first part of this thesis, we propose a technique of factorizing certain classes of arithmetic functions and prove some results about factorization which are based mainly on the norms of such functions. The technique is a generalization of the original works of Rearick which consists of solving a special system of linear differential equations whose coefficients are polynomials in the function to be factorized. The solutions of this system are proved to be the sought after factors with increasing norms. Examples illustrating the technique are also given. In 1986, Shapiro and Sparer made an extensive study of algebraic independence of Dirichlet series. Since the ring of Dirichlet series is isomorphic to that of arithmetic functions, the study in one setting is then equivalent to the other. Shapiro and Sparer's investigation began with a theorem asserting that Dirichlet series are algebraically independent if their Jacobian does not vanish, which is classical in the case of real-valued functions. Taking the Riemann zeta function as a building block, they discovered that Dirichlet series algebraically dependent on the zeta function can uniquely be represented as power series in the logarithms of zeta function. A number of algebraic dependence results of Dirichlet series were derived as consequences. Shapiro and Sparer then went on to investigate analogous results for formal generalized Dirichlet series. Results in the second part of this thesis either extend or simplify some of Shapiro and Sperer's results. These include, for example, replacing the zeta function by Dirichlet series with completely multiplicative coefficients to obtain similar log-series expansions, dependence of series with infinite support, and dependence of non-units whose norms are relatively prime.

In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π 2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζ G (2) where ζ G is the zeta function associated with the Gaussian integers. In order to calculate the Gaussian probability, we use a theorem that enables us to factor the zeta function into a product of the Riemann zeta function and a Dirichlet series called an L-series. For the Gaussian integers we get: ζ G (2)=ζ(2)L(2,χ), where $$L\left( {2,\chi } \right) = 1 - \frac{1}{{3^2 }} + \frac{1}{{5^2 }} - \frac{1}{{7^2 }} + \cdots .$$ We use this factorization to approximate the Gaussian probability to 17 decimal places.

The property of zeta functions on mixed joint universality in the Voronin’s sense states that any two holomorphic functions can be approximated simultaneously with an accuracy of ε&gt;0 by suitable vertical shifts of the pair consisting the Riemann and Hurwitz zeta functions. A rather general result can be obtained for the classes of zeta functions, particularly when an approximating pair is composed of the Matsumoto zeta functions’ class and the periodic Hurwitz zeta function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably ε&gt;0. Moreover, we provide concluding remarks on certain more general mixed tuples of zeta functions.

This article explores the connection between Fourier series and various zeta functions, including the Riemann zeta function and its generalizations. Specifically, we derive recurrence formulas for even and odd values of zeta functions using Fourier expansions, extending these results to the Hurwitz zeta function.

This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s=1, and further that it has a zero at all non-positive even integers, i.e. at s=0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.