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For 0 ≤ α ≤ 1 , let L α (x) = ∑ n ≤ x λ (n) n α , where λ (n) is the Liouville function. Then famous criteria of Pólya and Turán claim that the eventual sign constancy of each of L 0 (x) and L 1 (x) alone implies the Riemann hypothesis. However, Haselgrove disproved the eventual sign constancy hypothesis. As a remedy for this, we show that the eventual negativity of each of L α (x) , L α ′ (x) , L α ″ (x) for all 1 / 2 < α < 1 is equivalent to the Riemann hypothesis. Analogous equivalent conditions for the Riemann hypothesis are formulated in terms of partial sums of the Möbius function as well. Our criteria indicate a new tendency concerning the values of the Liouville and Möbius functions in stark contrast with their random behavior. Further generalizations are given in two essential directions, namely that equivalent criteria are developed for any quasi-Riemann hypothesis, and we allow the partial sums to be supported on semigroups after sifting out integers divisible by members of a set of prime numbers whose size is under control. Our negativity conditions for the quasi-Riemann hypothesis exhibit a curious rigidity in two respects, first it doesn't matter which particular primes we use to sift out, but only the size of the set, second the sign of the partial sums is unaltered with respect to two successive derivatives. Finally, other competing necessary or sufficient conditions for a quasi-Riemann hypothesis are studied regarding the monotonicity, zeros and sign changes of partial sums of Dirichlet series. [ABSTRACT FROM AUTHOR]

Inspired by representations of the class number of imaginary quadratic fields, in this paper, we give explicit evaluations of trigonometric series having generalized harmonic numbers as coefficients in terms of odd values of the Riemann zeta function and special values of L -functions subject to the parity obstruction. The coefficients that arise in these evaluations are shown to belong to certain cyclotomic extensions. Furthermore, using best polynomial approximation of smooth functions under uniform convergence due to Jackson and their log-sine integrals, we provide approximations of real numbers by combinations of special values of L -functions corresponding to the Legendre symbol. Our method for obtaining these results rests on a careful study of generating functions on the unit circle involving generalized harmonic numbers and the Legendre symbol, thereby relating them to values of polylogarithms and then finally extracting Fourier series of special functions that can be expressed in terms of Clausen functions. [ABSTRACT FROM AUTHOR]

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramér on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem. [ABSTRACT FROM AUTHOR]

The analytic continuation of a family of Dirichlet series whose coefficients are partition functions having parts in a finite set is established. The singularities arising from this continuation are shown to be finitely many simple poles, where the residues are computable positive rational numbers. Then we give several applications. First, special values of the continuation are studied in terms of algebraic combinations of the Hurwitz zeta function. Second, a transformation formula is obtained for exponentially decaying series involving these partition functions using special values of the continuation at negative integers. Finally, asymptotic behavior and orthogonality of certain convolutions with the normalization of these partition functions are investigated in the mean value sense with precise error terms. [ABSTRACT FROM AUTHOR]

Abstract: Sign of averages of Ramanujan sums is studied and it is shown that these averages have a curious tendency to be positive. This in turn gives that Ramanujan sums are nearly orthogonal to a family of vectors whose entries are powers of consecutive integers. Further applications are given to the limit points of these averages along semigroups of integers, the peak size of partial sums of Ramanujan sum and an optimization problem on weighted exponential sums supported on reduced residue systems. Exact evaluations of trigonometric sums having combinatorially significant coefficients and subject to divisibility constraints are obtained in terms of Bernoulli numbers. [Copyright &y& Elsevier]

Abstract: Using an integral transform with a mild singularity, we obtain series representations valid for specific regions in the complex plane involving trigonometric functions and the central binomial coefficient which are analogues of the types of series representations first studied by Ramanujan over certain intervals on the real line. We then study an exponential type series rapidly converging to the special values of L-functions and the Riemann zeta function. In this way, a new series converging to Catalanʼs constant with geometric rate of convergence less than a quarter is deduced. Further evaluations of some series involving hyperbolic functions are also given. [Copyright &y& Elsevier]

Abstract: We obtain quantitative upper bounds on partial sums of the Möbius function over semigroups of integers in an arithmetic progression. Exploiting the cancellation of such sums, we deduce upper bounds for the discrepancy of fractions in the unit interval whose denominators satisfy the same restrictions. In particular, the uniform distribution and approximation of discrete weighted averages over such fractions are established as a consequence. [Copyright &y& Elsevier]

Abstract: Let χ be a Dirichlet character and be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordanʼs and Eulerʼs totient function for the mean square average of when χ ranges over all odd characters modulo k and 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 if χ and have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when . Consequently, we see that for almost all odd characters modulo k, , where is any function monotonically tending to infinity. [Copyright &y& Elsevier]

Abstract: We prove the existence of the pair correlation measure associated to torsion points on the real locus of an elliptic curve E and provide an explicit formula for the limiting pair correlation function. [Copyright &y& Elsevier]

Abstract: We investigate local spacing problems along curves via smooth maps. Moreover, we provide explicit formulas for the nearest neighbor spacing distribution function of torsion points on elliptic curves over , and of rational points on the unit circle. [Copyright &y& Elsevier]

Abstract: Given an irrational number α and a sequence B of coprime positive integers with the sum of inverses convergent, we investigate the problem of finding small values of nα , with n B-free. [Copyright &y& Elsevier]

Abstract: Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions. [Copyright &y& Elsevier]

Let a(n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a(n). More generally, we get short interval version of results of Ivic´ on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdo¨s and Szekeres on the summatory function of a(n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups. [Copyright &y& Elsevier]