This paper aims to study the Betti homology and de Rham cohomology of twisted symmetric powers of the Kloosterman connection of rank two on the torus. We compute the period pairing and, with respect to certain bases, interpret these associated period numbers in terms of the Bessel moments. Via the rational structures on Betti homology and de Rham cohomology, we prove the (mathbb {Q})-linear and quadratic relations among these Bessel moments.
We present a direct proof of an inversion formula for a hyperelliptic integral that is not recorded by Ramanujan in his second notebook.
In 1974, Carlitz and Scoville introduced the Stirling–Eulerian polynomial (A_n(x,y|alpha ,beta )) as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of (A_n(x,y|alpha ,beta )), denoted (P_n(u_1,u_2,u_3,u_4|alpha ,beta )), which is the enumerator of permutations by valleys, peaks, double ascents, double descents, left-to-right maxima and right-to-left maxima. Using Chen’s context-free grammar calculus, Ji proved a formula for the generating function of (P_n(u_1,u_2,u_3,u_4|alpha ,beta )), generalizing the work of Carlitz and Scoville. Ji’s formula has many nice consequences, one of which is an intriguing (gamma )-positivity expansion for (A_n(x,y|alpha ,beta )). In this paper, we prove a q-analog of Ji’s formula by using Gessel’s q-compositional formula and provide a combinatorial approach to her (gamma )-positivity expansion of (A_n(x,y|alpha ,beta )).
Let (textrm{C}_{n}) and (textrm{Q}_{n}) denote the cyclic group and the generalized quaternion group of order n, respectively. We determine all possible values of the integer group determinants of (textrm{C}_{8} rtimes _{3} textrm{C}_{2}) and (textrm{Q}_{8} rtimes textrm{C}_{2}), which are the unresolved groups of order 16 (Serrano, Paudel and Pinner also obtained a complete description of the integer group determinants of (textrm{Q}_{8} rtimes textrm{C}_{2}) independently of this paper and presented it a few days earlier than this paper). Also, we give a diagram of the set inclusion relations between the integer group determinants for all groups of order 16
Matomäki proved that if (alpha in {mathbb {R}}) is irrational, then there are infinitely many primes p such that (|alpha -a/p|le p^{-4/3+varepsilon }) for a suitable integer a. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke L-functions.
Let (pequiv 1pmod {8}) and (qequiv 7pmod 8) be two prime numbers. The purpose of this paper is to compute the unit groups of the fields (mathbb {L}=mathbb {Q}(sqrt{2}, sqrt{p}, sqrt{q})) and give their 2-class numbers.
In this paper, we establish a new mean value theorem of Bombieri–Vinogradov’s type over Piatetski–Shapiro sequence. Namely, it is proved that for any given constant (A>0) and any sufficiently small (varepsilon >0), there holds
$$begin{aligned} sum _{begin{array}{c} dleqslant x^xi (d,l)=1 end{array}}Bigg |sum _{begin{array}{c} A_1(x)leqslant a<A_2(x) (a,d)=1 end{array}}g(a) Bigg (sum _{begin{array}{c} apleqslant x apequiv l!!!!!pmod d ap=[k^{1/gamma }] end{array}}1-frac{1}{varphi (d)}sum _{begin{array}{c} apleqslant x ap=[k^{1/gamma }] end{array}} 1Bigg )Bigg |ll frac{x^gamma }{(log x)^A}, end{aligned}$$provided that (1leqslant A_1(x)<A_2(x)leqslant x^{1-varepsilon }) and (g(a)ll tau _r^s(a)), where (lnot =0) is a fixed integer and
$$begin{aligned} xi :=xi (gamma )=frac{2^{38}+17}{38}gamma -frac{2^{38}-1}{38}-varepsilon end{aligned}$$with
$$begin{aligned} 1-frac{18}{2^{38}+17}<gamma <1. end{aligned}$$Moreover, for (gamma ) satisfying
$$begin{aligned} 1-frac{0.03208}{2^{38}+17}<gamma <1, end{aligned}$$we prove that there exist infinitely many primes p such that (p+2=mathcal {P}_2) with (mathcal {P}_2) being Piatetski–Shapiro almost–primes of type (gamma ), and there exist infinitely many Piatetski–Shapiro primes p of type (gamma ) such that (p+2=mathcal {P}_2). These results generalize the result of Pan and Ding [37] and constitute an improvement upon a series of previous results of [29, 31, 39, 47].
In this paper we study the modular differential equation (y''+s,E_4, y=0) where (E_4) is the weight 4 Eisenstein series and (s=pi ^2r^2) with (r=n/m) being a rational number in reduced form such that (mge 7). This study is carried out by solving the associated Schwarzian equation ({h,tau }=2,s,E_4) and using the theory of equivariant functions on the upper half-plane and the 2-dimensional vector-valued modular forms. The solutions are expressed in terms of the Gauss hypergeometric series. This completes the study of the above-mentioned modular differential equation of the associated Schwarzian equation given that the cases (1le mle 6) have already been treated in Saber and Sebbar (Forum Math 32(6):1621–1636, 2020; Ramanujan J 57(2):551–568, 2022; J Math Anal Appl 508:125887, 2022; Modular differential equations and algebraic systems, http://arxiv.org/abs/2302.13459).
We study the square values of Littlewood polynomials. Using various methods we give all these values for the degrees (n=3, 5) and (nle 24) even. Beside this, we gather computational data (by providing all solutions in a certain range) for n odd with (nle 17). We propose some striking problems for further research, as well.
Let (kge 3) be a positive integer and let (g(x)=a_{k}x^{k}+a_{k-1}x^{k-1}+cdots +a_0in mathbb {Z}[x]) with (gcd (a_{0}, ldots , a_{k-1},a_{k})=1, a_{k}>0). In this paper, we investigate the density of natural numbers which can be represented by the form (2^{g(j_1)}+2^{g(j_2)}+p), where (j_1,j_2) are positive integers and p is an odd prime.
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