The joint universality theorem for a pair of Hurwitz zeta functions

In this paper we investigate the joint functional distribution for a pair of Hurwitz zeta functions ζ(s,αj) (j=1,2) in the case that real transcendental numbers α₁ and α₂ satisfy α₂∈Q(α₁). Especially we establish the joint universality theorem for these zeta functions.     

Dirichlet series associated with polynomials and applications

We consider the possibility of the analytic continuation of the Dirichlet series SP;Z(s) associated with a polynomial P(x) and auxiliary series Z(s). In fact, we derive a certain criterion for the analytic continuation for some class of polynomials and give examples such that SP;Z(s) cannot be continued meromorphically to the whole plane C. We also study the asymptotic behaviors of the sum MP(x)=_∑P(n₁,…,nk)?xΛ(n₁)?Λ(nk) considered first by M. Peter. Generalizations of this sum are also considered.     

New results on the Stieltjes constants: Asymptotic and exact evaluation

The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s=1. We present new asymptotic, summatory, and other exact expressions for these and related constants.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

Newman's phenomenon for generalized Thue-Morse sequences

Let t_j=(-1)^s(j) be the Thue-Morse sequence with s(j) denoting the sum of the digits in the binary expansion of j. A well-known result of Newman [On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721] says that t₀+t₃+t₆+?+t_3k>0 for all k?0.In the first part of the paper we show that t₁+t₄+t₇+?+t_3k+1<0 and t₂+t₅+t₈+?+t_3k+2?0 for k?0, where equality is characterized by means of an automaton. This sharpens results given by Dumont [Discrépance des progressions arithmétiques dans la suite de Morse, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 145-148]. In the second part we study more general settings. For a,g?2 let ω_a=exp(2πi/a) and , where s_g(j) denotes the sum of digits in the g-ary digit expansion of j. We observe trivial Newman-like phenomena whenever a|(g-1). Furthermore, we show that the case a=2 inherits many Newman-like phenomena for every even g?2 and large classes of arithmetic progressions of indices. This, in particular, extends results by Drmota and Ska?ba [Rarified sums of the Thue-Morse sequence, Trans. Amer. Math. Soc. 352 (2000) 609-642] to the general g-case.     

Beurling primes with large oscillation

A Beurling generalized number system is constructed having integer counting function N_B(x) = κx +O(x^θ) with κ>0 and 1/2 <θ <1, whose prime counting function satisfies the oscillation estimate π_B(x) =li(x) + Ω(xexp(-c)), and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ). The construction uses elements of classical analytic number theory and probability. The author was supported in part by NSF grants DMS-0070720 and DMS-0244660. The author was supported in part by NSF grant DMS-0244660.     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

The existence and the non-existence of joint t-universality for Lerch zeta functions

In this paper, we show the following theorems. Suppose 0<al<1 are algebraically independent numbers and 0<λl?1 for 1?l?m. Then we have the joint t-universality for Lerch zeta functions L(λl,al,s) for 1?l?m. Next we generalize Lerch zeta functions, and obtain the joint t-universality for them. In addition, we show examples of the non-existence of the joint t-universality for Lerch zeta functions and generalized Lerch zeta functions.     

Integral complete multipartite graphs

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphs K_p1,p2,…,pr=K_{a1·p1,a2·p2,…,as·ps} with s=3,4. We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s=4, we give a positive answer to a question of Wang et al. [Integral complete r-partite graphs, Discrete Math. 283 (2004) 231-241]. The problem of the existence of integral complete multipartite graphs K_{a1·p1,a2·p2,…,as·ps} with arbitrarily large number s remains open.     

The Riemann hypothesis for certain integrals of Eisenstein series

This paper studies the nonholomorphic Eisenstein series E(z,s) for the modular surface PSL(2,Z)\H, and shows that integration with respect to certain nonnegative measures μ(z) gives meromorphic functions Fμ(s) that have all their zeros on the line . For the constant term a₀(y,s) of the Eisenstein series the Riemann hypothesis holds for all values y?1, with at most two exceptional real zeros, which occur exactly for those y>4πe−γ=7.0555+. The Riemann hypothesis holds for all truncation integrals with truncation parameter T?1. At the value T=1 this proves the Riemann hypothesis for a zeta function Z2,Q(s) recently introduced by Lin Weng, associated to rank 2 semistable lattices over Q.     

Dirichlet series related to the Riemann zeta function

A study is made of the function H(s, z) defined by analytic continuation of the Dirichlet series H(s, z) = Σ_n=1^∞n^?sΣ_m=1ⁿm^?z, where s and z are complex variables. For each fixed z it is shown that H(s, z) exists in the entire s-plane as a meromorphic function of s, and its poles and residues are determined. Also, for each fixed s ≠ 1 it is shown that H(s, z) exists in the entire z-plane as a meromorphic function of z, and again its poles and residues are determined. Two different representations of H(s, z) are given from which a reciprocity law, H(s, z) + H(z, s) = ζ(s) ζ(z) + ζ(s + z), is deduced. For each integer q ≥ 0 the function values H(s, ?q) and H(?q, s) are expressed in terms of the Riemann zeta function. Similar results are also obtained for the Dirichlet series T(s, z) = Σ_n=1^∞n^?sΣ_m=1ⁿm^?z (m + n)^?1. Applications include identities previously obtained by Ramanujan, Williams, and Rao and Sarma.     

Twisted Hecke L-values and period polynomials

Let f₁,…,fd be an orthogonal basis for the space of cusp forms of even weight 2k on Γ₀(N). Let L(fi,s) and L(fi,χ,s) denote the L-function of fi and its twist by a Dirichlet character χ, respectively. In this note, we obtain a “trace formula” for the values at integers m and n with 0<m,n<2k and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisely the value of the ratio L(f,χ,m)/L(f,n) for a Hecke eigenform f.     

On the mean square average of special values of L-functions

Let χ be a Dirichlet character and L(s,χ) be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordan?s and Euler?s totient function for the mean square average of L(1,χ) when χ ranges over all odd characters modulo k and L(2,χ) when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of L(r,χ) if χ and r?1 have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when r?3. Consequently, we see that for almost all odd characters modulo k, |L(1,χ)|<Φ(k), where Φ(x) is any function monotonically tending to infinity.     

Euler numbers congruences and Dirichlet L-functions

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In this paper we apply Yamamoto's Theorem [Y. Yamamoto, Dirichlet series with periodic coefficients, in: Proc. Intern. Sympos. “Algebraic Number Theory”, Kyoto, 1976, JSPS, Tokyo, 1977, pp. 275-289] to find the residue modulo a prime power of the linear combination of Dirichlet L-function values L(s,χ) at positive integral arguments s such that s and χ are of the same parity, in terms of Euler numbers, whereby we obtain the finite expressions for short interval character sums. The results obtained generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s=1.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=_KAv4FCdVUs.     

A Tauberian theorem related to Weyl's criterion

Let {x_n} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σ_n=1^Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t₀ for which A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞. In case f(s) = Σ_n=1^∞a(n)e^?sx_n, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)^?1f(σ + it₀) → 0 as σ → 0 + if and only if A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞.     

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

We define the multiple zeta function of the free Abelian group Z^d as

ζ_Z^d(s₁,…,s_d)=∑_|Z^d:H|<∞α₁(H)^−s₁?α_d(H)^−s_d,     

Partitions of the set of natural numbers and their representation functions

For a given set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. In this paper we give a simple proof to two results by Sándor.     

Existence of positive solution for some class of nonlinear fractional differential equations

In this paper, we investigate the multiple and infinitely solvability of positive solutions for nonlinear fractional differential equation Du(t)=t^νf(u), 0<t<1, where D=t^−βδD_β^γ−δ,δ, β>0, γ?0, 0<δ<1, ν>−β(γ+1). Our main work is to deal with limit case of f(s)/s as s→0 or s→∞ and Φ(s)/s, Ψ(s)/s as s→0 or s→∞, where Φ(s), Ψ(s) are functions connected with function f. In J. Math. Appl. 252 (2000) 804-812, we consider the existence of a positive solution for the particular case of Eq. (1.1), i.e., the Riemann-Liouville type (β=1, γ=0) nonlinear fractional differential equation, using the super-lower solutions method. Here, we devote to the existence of positive solution and multi-positive solutions for Eq. (1.1) by means of the fixed point theorems for the cone.     

Müntz formula and zero free regions for the Riemann zeta function

A formula first derived by Müntz which relates the Riemann zeta function ζ times the Mellin transform of a test function f and the Mellin transform of the theta transform of f is exploited, together with other analytic techniques, to construct zero free regions for ζ(s) with s in the critical strip. Among these are regions with a shape independent of Res.     

Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a²+b²+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a²+b²=p.