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Let p be a prime number. For each natural number n, we study the behavior of the function fp(n) which enumerates the number of factorizations ab = n with a + b a perfect square (mod p). The study of this function is inspired by the cognate function f(n) which enumerates the number of factorizations ab = n with a + b a perfect square. The descent theory of elliptic curves would show that if f(n) is unbounded for squarefree values of n, then there are elliptic curves over the rational number field with arbitrarily large rank. In this note, we show for every prime p, fp(n) is unbounded as n ranges over squarefree values, thus providing some evidence for the conjecture that f(n) is unbounded for squarefree n. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over \mathbb {Q} with discriminant \Delta _E. For primes p of good reduction, let N_p be the number of points modulo p and write N_p=p+1-a_p. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies \begin{equation*} \lim _{x\to \infty }\frac {1}{\log x}\sum _{\substack {p\leq x\\ p\nmid \Delta _{E}}}\frac {a_p\log p}{p}=-r+\frac {1}{2}, \end{equation*} where r is the order of the zero of the L-function L_{E}(s) of E at s=1, which is predicted to be the Mordell-Weil rank of E(\mathbb {Q}). We show that if the above limit exits, then the limit equals -r+1/2. We also relate this to Nagao's conjecture. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit. [ABSTRACT FROM AUTHOR]

We study the normal number of prime factors of a f (p) + a g (p) with p prime and f , g distinct Hecke eigenforms of weight two. Assuming a quasi-generalized Riemann hypothesis, we show that the normal order is log log p. We also obtain an estimate for the number of primes p for which a f (p) + a g (p) = 0. [ABSTRACT FROM AUTHOR]

For an algebraic number field K , let ζ K (s) be the associated Dedekind zeta-function. It is conjectured that ζ K (m) is transcendental for any positive integer m > 1. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when K is a totally real number field, ζ K (2 n) is an algebraic multiple of π 2 n [ K : ℚ ] and hence, is transcendental. If K is not totally real, the question of whether ζ K (m) is irrational or not remains open. In this paper, we prove that for a fixed integer n ≥ 1 , at most one of ζ K (2 n + 1) is rational, as K varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields. [ABSTRACT FROM AUTHOR]

We investigate generalizations arising from the identity ζ 2 (n − 1 , 1) = n − 1 2 ζ (n) − 1 2 ∑ j = 2 n − 2 ζ (j) ζ (n − j) , where ζ 2 (k , 1) denotes a double zeta value at (k , 1) , or an Euler-Zagier sum. In particular, we prove analogues of the above identity for Lerch zeta-functions and Dirichlet L -functions. Such an attempt has met with limited success in the past. We highlight that this study naturally leads one into the realm of multiple L -values and multiple L ⁎ -values. [ABSTRACT FROM AUTHOR]

We prove central limit theorems (under suitable growth conditions) for sums of quadratic characters, families of Hecke eigenforms of level 1 and weight k, and families of elliptic curves, twisted by an L-function satisfying certain properties. As a corollary, we obtain a central limit theorem for products χ (p)af(p) where χ is a quadratic Dirichlet character and f is a normalized Hecke eigenform. [ABSTRACT FROM AUTHOR]

Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form a n − b n , denoted P (a n − b n) , by essentially proving that for integers a > b > 0 , P (a n − b n) ≥ n + 1 for every n > 2. Since then, the problem of finding bounds on the largest prime factor of Lehmer sequences, Lucas sequences or special cases thereof has been studied by many, most notably by Schinzel (1962), and Stewart (1975, 2013). In 2002, Murty and Wong proved, conditionally upon the abc conjecture, that P (a n − b n) ≫ n 2 − ϵ for any ϵ > 0. In this article, we improve this result for the specific case where b = 1. Specifically, we obtain a more precise result, and one that is dependent on a condition we believe to be weaker than the abc conjecture. Our result actually concerns the largest prime factor of the n th cyclotomic polynomial evaluated at a fixed integer a , P (Φ n (a)) , as we let n grow. We additionally prove some results related to the prime factorization of Φ n (a). We also present a connection to Wieferich primes, as well as show that the finiteness of a particular subset of Wieferich primes is a sufficient condition for the infinitude of non-Wieferich primes. Finally, we use the technique used in the proof of the aforementioned results to show an improvement on average of estimates due to Erdős for certain sums. [ABSTRACT FROM AUTHOR]

We establish an error term in the Sato–Tate theorem of Birch. That is, for $p$ prime, $q=p^{r}$ and an elliptic curve $E:y^{2}=x^{3}+ax+b$ , we show that $$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$ for any interval $I\subseteq [0,\unicode[STIX]{x1D70B}]$ , where the quantity $\unicode[STIX]{x1D703}_{a,b}$ is defined by $2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$ and $\unicode[STIX]{x1D707}_{ST}(I)$ denotes the Sato–Tate measure of the interval $I$. [ABSTRACT FROM AUTHOR]

Abstract In 1927, Artin conjectured that any integer other than −1 or a perfect square generates the multiplicative group (Z / p Z) × for infinitely many p. In 2000, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture. In this article, we prove an unconditional lower bound for this two-variable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the first N terms of a non-degenerate binary recurrence sequence. We also prove a weaker version of the same theorem, and give three proofs that we consider to be of independent interest. The first proof uses a transcendence result of Stewart, the second uses a theorem of Bombieri and Schmidt on Thue equations and the third uses Mumford's gap principle for counting points on curves by their height. We finally prove a disjunction theorem, where we consider the set of primes satisfying either our two-variable condition or the original condition of Artin's conjecture. We give an unconditional lower bound for the number of such primes. [ABSTRACT FROM AUTHOR]

In this paper, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms f and g with Fourier coefficients a n and b n respectively, we consider the following questions: existence of infinitely many primes p such that a p b p ≠ 0 ; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals. [ABSTRACT FROM AUTHOR]

We use recent advances in sieve theory to show that conditional upon the generalized Elliott-Halberstam conjecture, at least one of the following is true: (i) Artin's primitive root conjecture holds for all a not equal to ±1 or a perfect square (ii) The Lang-Trotter conjecture holds for all CM elliptic curves E/Q with rank E(Q) ≥ 1 and CM field k ≠ Q(ω),Q(i), where ω is a primitive cube root of unity. [ABSTRACT FROM AUTHOR]

Hilbert's tenth problem for rings of integers of number fields remains open in general, although a negative solution has been obtained by Mazur and Rubin conditional to a conjecture on Shafarevich–Tate groups. In this work we consider the problem from the point of view of analytic aspects of L -functions instead. We show that Hilbert's tenth problem for rings of integers of number fields is unsolvable, conditional to the following conjectures for L -functions of elliptic curves: the automorphy conjecture and the rank part of the Birch and Swinnerton–Dyer conjecture. [ABSTRACT FROM AUTHOR]

For two arithmetical functions f and g , we study the convolution sum of the form ∑ n ≤ N f ( n ) g ( n + h ) in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup. [ABSTRACT FROM AUTHOR]

Let f1, f2 be two distinct normalized Hecke eigenforms of weights k1 and k2 with at least one of them not of CM type and with p-th Hecke eigenvalues given by ap(f1)p(k1-1)/2 and ap(f2)p(k2-1)/2 respectively and p being prime. If ap(f1) = ap(f2) for a set of primes with positive upper density, then we show that f1 = f2 ⊗ Χ for some Dirichlet character Χ. [ABSTRACT FROM AUTHOR]

For m ≥ 2, let π be an irreducible cuspidal automorphic representation of GLm(AQ) with unitary central character. Let aπ(n) be the nth coefficient of the L-function attached to π. Goldfeld and Sengupta have recently obtained a bound for Σn≤x aπ(n) as x → ∞. For m ≥ 3 and π not a symmetric power of a GL2(AQ)-cuspidal automorphic representation with not all finite primes unramified for π, their bound is better than all previous bounds. In this paper, we further improve the bound of Goldfeld and Sengupta. We also prove a quantitative result for the number of sign changes of the coefficients of certain automorphic L-functions, provided the coefficients are real numbers. [ABSTRACT FROM AUTHOR]

In this article we deal with the problem of counting the number of pairs of normalized eigenforms (f, g) of weight k and level N such that ap(f) = ap(g) where ap(f) denotes the p-th Fourier coefficient of f. Here p is a fixed prime. [ABSTRACT FROM AUTHOR]

In 2011, M.R. Murty and V.K. Murty [10] proved that if L ( s , χ D ) is the Dirichlet L -series attached a quadratic character χ D , and L ′ ( 1 , χ D ) = 0 , then e γ is transcendental. This paper investigates such phenomena in wider collections of L -functions, with a special emphasis on Artin L -functions. Instead of s = 1 , we consider s = 1 / 2 . More precisely, we prove that exp ⁡ ( L ′ ( 1 / 2 , χ ) L ( 1 / 2 , χ ) − α γ ) is transcendental with some rational number α . In particular, if we have L ( 1 / 2 , χ ) ≠ 0 and L ′ ( 1 / 2 , χ ) = 0 for some Artin L -series, we deduce the transcendence of e γ . [ABSTRACT FROM AUTHOR]

For two arithmetical functions f and g with absolutely convergent Ramanujan expansions, Murty and Saha have recently derived asymptotic formulas with error term for the convolution sum ∑ n ≤ N f ( n ) g ( n + h ) under some suitable conditions. In this follow up article we improve these results with a weakened hypothesis which is in some sense minimal. [ABSTRACT FROM AUTHOR]

We extend Euler's celebrated theorem evaluating ζ(2k). We replace the terms n-2k in the infinite sum for ζ(2k), with (n2 + Bn +C)-k where B,C are complex and k is a positive integer. We explicitly evaluate these sums and also briefly discuss their transcendence [ABSTRACT FROM AUTHOR]

Given two arithmetical functions f , g we derive, under suitable conditions, asymptotic formulas with error term, for the convolution sums ∑ n ≤ N f ( n ) g ( n + h ) , building on an earlier work of Gadiyar, Murty and Padma. A key role in our method is played by the theory of Ramanujan expansions for arithmetical functions. [ABSTRACT FROM AUTHOR]

Klein's modular j-function is defined to be *** where z ∈ Ć with ***(z) > 0, q = exp(2iπz), E4(z) denotes the normalized Eisenstein series of weight 4 and Δ(z) is the Ramanujan's Delta function. In this short note, we show that for each integer a > 1, the interval (a, 4a(a + 1)) (respectively, the interval (16a -- 1, (4a + 1)2)) contains an integer n with n ≡ 7 (mod 8) such that c(n) is odd (respectively, c(n) is even). [ABSTRACT FROM AUTHOR]

For any periodic function f : N → C with period q , we study the Dirichlet series L ( s , f ) : = ∑ n ≥ 1 f ( n ) / n s . It is well-known that this admits an analytic continuation to the entire complex plane except at s = 1 , where it has a simple pole with residue ρ : = q − 1 ∑ 1 ≤ a ≤ q f ( a ) . Thus, the function is analytic at s = 1 when ρ = 0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L -series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L ( 1 , f ) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L ( 1 , f ) . [ABSTRACT FROM AUTHOR]

It was shown by Granville that the ABC conjecture allows one to prove asymptotic estimates on the number of squarefree values of polynomials. However, his proof gives no information on the error term of the asymptotic formula. On the ABC conjecture, we prove an asymptotic formula with error term using a different technique. From the ABC conjecture we also deduce an asymptotic formula with error term for the number of squarefree values of polynomials on certain sets of integers that are residually well distributed in a suitable sense. [ABSTRACT FROM AUTHOR]

Let M ∈ F q [ t ] be a polynomial, and let k ≥ 2 be an integer. In this note, we will compute the asymptotic density of irreducible monic polynomials P ∈ F q [ t ] for which P + M is not divisible by the k th power of any irreducible polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we link the nature of special values of certain Dirichlet L-functions to those of multiple gamma values. [ABSTRACT FROM AUTHOR]

Let d(n) denote the number of divisors of n. In this paper, we study the average value of d(a(p)), where p is a prime and a(p) is the p-th Fourier coefficient of a normalized Hecke eigenform of weight k ≥ 2 for Γ0(N) having rational integer Fourier coefficients. [ABSTRACT FROM AUTHOR]

Abstract: After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Freyʼs height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Freyʼs height conjecture which applies to all elliptic curves over , not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of ), with a completely explicit and effective bound, without using any variant of Bakerʼs theory or the Thue–Bombieri method. [Copyright &y& Elsevier]

We will prove that if the unit rank of a number field with cyclic class group is large enough and if the Galois group of its Hilbert class field over ℚ is abelian, then every generator of its class group is a Euclidean ideal class. We use this to prove the existence of a non-principal Euclidean ideal class that is not norm-Euclidean by showing that ℚ(√5,√21,√22) has such an ideal class. [ABSTRACT FROM AUTHOR]

The classical theorem of Bombieri and Vinogradov is generalized to a non-abelian, non-Galois setting. This leads to a prime number theorem of "mixed-type" for arithmetic progressions "twisted" by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the Bombieri-Vinogradov theorem. We develop this theory with a view to applications in the study of the Euclidean algorithm in number fields and arithmetic orbifolds. [ABSTRACT FROM AUTHOR]

The article offers information on the proof of a Russian mathematician Pafnuty Lvovich Chebyshev's postulate. It mentions that an Indian mathematician S. Ramanujan used the Stirling series which relies on the properties of the gamma function to give a simpler proof. It also states that purpose of a calculus-free derivation of the series is not proved and thus must be removed from his approach.

We re-prove the Hardy-Ramanujan asymptotic formula for the partition function without using the circle method. We derive our result from recent work of Bruinier and Ono on harmonic weak Maass forms. [ABSTRACT FROM AUTHOR]

We obtain a new proof of an asymptotic formula for the coefficients of the j-invariant of elliptic curves. Our proof does not use the circle method. We use Laplace's method of steepest descent and the Hardy-Ramanujan asymptotic formula for the partition function. (The latter asymptotic formula can be derived without the circle method.) [ABSTRACT FROM AUTHOR]

The article focuses on Euler's numbers ɣ and a class of its analogues. It reports that numbers can be divided as either transcendental or algebraic and highlights that transcendental numbers were first discovered by Liouville's theorem and points out that three numbers π, e and Euler's ɣ are referred to as the holy trinity in Riemann hypothesis. It highlights that although the transcendence of π and e has been determined yet it has not been determined for Euler's numbers.

Let a be a natural number greater than 1. For each prime p, let ia(p) denote the index of the group generated by a in . Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko proved in 2004 that the average value of ia(p) is constant. We prove that the average value of ia(p) is constant without using Conjecture A of Hooley. More precisely, we show upon GRH that for any α with 0 ≤ α < 1, there is a positive constant cα > 0 such that where π(x) is the number of primes p ≤ x. We also study related questions. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field and f an integral ideal. Denote by Cl(f) the ray class group of f. For every non-trivial character χ of Cl(f), we show that L(1, χ)/π is transcendental. If f = f̄, then complex conjugation acts on the character group of Cl(f). Denoting by Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. the orbits of the group of characters, we show that the values L(1, χ) as χ ranges over elements of Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. are linearly independent over Q̄. We give applications of this result to the study of transcendental values of Petersson inner products and certain special values of Artin L-series attached to dihedral extensions. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite abelian group. If is a nonzero function with Fourier transform , the classical uncertainty principle states that . Recently, Tao showed that, if G is cyclic of prime order p, then in fact a stronger inequality holds. In this paper, we use representation theory of the unitary group and Weyl’s character formula to derive a generalization of Tao’s result for arbitrary finite cyclic groups. [Copyright &y& Elsevier]

We study the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al. The special values of such integrals at algebraic points in the upper half-plane are linked to Riemann zeta values at odd positive integers. [ABSTRACT FROM PUBLISHER]

The article provides mathematical analysis to a problem proposed by Harry Ruderman. It proves that the Ruderman's problem contains only finitely many pairs of natural numbers m and n. It explores how a version of the abc conjecture can be a solution to come up with the effective bounds on the mathematical constants m and n, since the proof involves Schmidt subspace theorem.

Abstract: Text: Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato–Tate law. We present two methods of proof. Both use the framework of Murty and Sinha (2009) ; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato–Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be done. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=faW2iDpe5IE. [ABSTRACT FROM AUTHOR]

Abstract: The Euler–Lehmer constants are defined as the limits We show that at most one number in the infinite list is an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdös. If is such that and , then Erdös conjectured that If , we show that the Erdös conjecture is true. [Copyright &y& Elsevier]

Abstract: We study transcendental values of the logarithm of the gamma function. For instance, we show that for any rational number x with , the number is transcendental with at most one possible exception. Assuming Schanuel''s conjecture, this possible exception can be ruled out. Further, we derive a variety of results on the Γ-function as well as the transcendence of certain series of the form , where and are polynomials with algebraic coefficients. [Copyright &y& Elsevier]

Abstract: In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space of cusp forms of weight k and level N. In this paper, we derive an effective version of Serre''s theorem. As a consequence, we estimate, for a given d and prime p coprime to N, the number of eigenvalues of the pth Hecke operator acting on of degree less than or equal to d. This allows us to determine an effectively computable constant such that if is isogenous to a product of -simple abelian varieties of dimensions less than or equal to d, then . We also study the effective equidistribution of eigenvalues of Frobenius acting on a family of curves over a fixed finite field as well as the eigenvalue distribution of adjacency matrices of families of regular graphs. These results are derived from a general “all-purpose” equidistribution theorem. [Copyright &y& Elsevier]

The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d – 1 is a prime power. In this paper, we consider the case when d – 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost” Ramanujan graphs for almost every value of d. More precisely, for any fixed ℇ > 0 and for almost every value of d (in the sense of natural density), there are infinitely many d-regular graphs such that all the non-trivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ℇ) . 2000 Mathematics Subject Classification: 05C50, 15A18, 11N05, 11F30. [ABSTRACT FROM AUTHOR]

The article presents a proof of Pick's theorem, using Minkowski's convex body theorem. It reports that Pick's theorem provided a formula for easily calculating the area of a planar polygon P, also called lattice polygon. It mentions that Pick's theorem states that if I is the number of lattice points in the interior of P and B is the number of lattice points on its boundary, then the Area of P is given by A = I + B―2 - 1. A review of Minkowski's convex body theorem is presented, where in a bounded, symmetric, convex region C is Rn with volume greater than 2n contains at least one nonzero lattice point. It suggests that Minkowski's theorem is true for spaces of any dimension, which might be able to generalize Pick's theorem to higher dimensions.

For a fixed hyperelliptic curve C given by the equation y2 = f( x) with f ∈ ℤ[ x] having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm whose equation is given by my2 = f( x). More precisely, we study the Dirichlet series where the summation is over all non-zero squarefree integers. We show that converges for ℜ( s) > 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in ℤ[ x]. [ABSTRACT FROM AUTHOR]

Abstract: We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of where is a prime and a polynomial with integer coefficients. [Copyright &y& Elsevier]

Abstract: We develop the Turán sieve and a ‘simple sieve’ in the context of bipartite graphs and apply them to various problems in combinatorics. More precisely, we provide applications in the cases of characters of abelian groups, vertex-colourings of graphs, Latin squares, connected graphs, and generators of groups. In addition, we give a spectral interpretation of the Turán sieve. [Copyright &y& Elsevier]

Examines the similarity between prime numbers and irreducible polynomials. Conjectures indicating their similarity; Theorems and proofs.

Let 2(n) denote the Liouville function and consider the sum S(x,α) σn≤x [supi(n)e[sup2πin&aalpha;]We prove that for all &aalpha; of irrational type 1 (that includes all algebraic irrationalities) S(x, &aalpha;) O(x[sup4/5, &brvbar&epsilon) for any ε > 0. The method is extended to study more general sums of the form σn≥x[supa,(n),e]²[suppi;inα] for a general class of arithmetical functions a(n). The main technique is the Vinogradov Vaughan method of studying exponential sums. [ABSTRACT FROM AUTHOR]