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For given positive integers m and n with m

We prove that the Riemann zeta-function ζ (σ + i t) has no zeros in the region σ ≥ 1 - 1 / (55.241 (log | t |) 2 / 3 (log log | t |) 1 / 3 ) for | t | ≥ 3 . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region σ ≥ 1 - 1 / (5.558691 log | t |) for | t | ≥ 2 . We also provide new bounds that are useful for intermediate values of | t | . Combined, our results improve the largest known zero-free region within the critical strip for 3 · 10 12 ≤ | t | ≤ exp (64.1) and | t | ≥ exp (1000) . [ABSTRACT FROM AUTHOR]

Let \digamma (n) denote a multiplicative function with range \{-1,0,1\}, and let F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \digamma (n). Then F(x)/\sqrt {x} = a\sqrt {x} + b + E(x), where a and b are constants and E(x) is an error term that either tends to 0 in the limit or is expected to oscillate about 0 in a roughly balanced manner. We say F(x) has persistent bias b (at the scale of \sqrt {x}) in the first case, and apparent bias b in the latter. For example, if \digamma (n)=\mu (n), the Möbius function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \mu (n) has apparent bias 0, while if \digamma (n)=\lambda (n), the Liouville function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \lambda (n) has apparent bias 1/\zeta (1/2). We study the bias when \digamma (p^k) is independent of the prime p, and call such functions fake \mu 's. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias. For such a function F(x) with apparent bias b, we also show that F(x)/\sqrt {x}-a\sqrt {x}-b changes sign infinitely often. [ABSTRACT FROM AUTHOR]

Let χ denote a primitive, non-quadratic Dirichlet character with conductor q , and let L (s , χ) denote its associated Dirichlet L -function. We show that | L (1 , χ) | ≥ 1 / (9.12255 log ⁡ (q / π)) for sufficiently large q , and that | L (1 , χ) | ≥ 1 / (9.69030 log ⁡ (q / π)) for all q ≥ 2 , improving some results of Louboutin. The improvements come from an averaging argument that simplifies Louboutin's approach, and the construction of certain trigonometric polynomials using simulated annealing, a technique often used in applied mathematics and computer science. We highlight the benefits of this approach to a pure mathematical problem. [ABSTRACT FROM AUTHOR]

We examine oscillations in a number of sums of arithmetic functions involving Ω (n) , the total number of prime factors of n , and ω (n) , the number of distinct prime factors of n. In particular, we examine oscillations in S α (x) = ∑ n ≤ x (− 1) n − Ω (n) / n α and in H α (x) = ∑ n ≤ x (− 1) ω (n) / n α for α ∈ [ 0 , 1 ] , and in W (x) = ∑ n ≤ x (− 2) Ω (n) . We show for example that each of the inequalities S 0 (x) < 0 , S 0 (x) > 3. 3 x , S 1 (x) > 0 , and S 1 (x) x < − 3. 3 is true infinitely often, disproving some hypotheses of Sun. [ABSTRACT FROM AUTHOR]

Let Rk(x) denote the error incurred by approximating the number of k-free integers less than x by x/\zeta (k). It is well known that Rk(x) = Ω (x1/2k), and widely conjectured that Rk(x) = O(x1/2k + ε). By establishing weak linear independence of some subsets of zeros of the Riemann zeta function, we establish an effective proof of the lower bound, with significantly larger bounds on the constant compared to those obtained in prior work. For example, we show that Rk(x)/x1/2k > 3 infinitely often and that Rk(x)/x1/2k < −3 infinitely often, for k = 2, 3, 4, and 5. We also investigate R2(x) and R3(x) in detail and establish that our bounds far exceed the oscillations exhibited by these functions over a long range: for 0 < x ≤ 1018 we show that |R2(x)| < 1.12543x1/4 and |R3(x)| < 1.27417x1/6. We also present some empirical results regarding gaps between square-free numbers and between cube-free numbers. [ABSTRACT FROM AUTHOR]

We establish some new congruences satisfied by the Lind Mahler measure on p-groups, and use them to determine the Lind-Lehmer constant for many finite groups. First, we determine the minimal nontrivial measure of pgroups where one component has particularly high order. Second, we describe an algorithm that determines a small set of possible values for the minimal nontrivial measure of a p-group of the form Zp × Zpk with k ≥ 2. This algorithm is remarkably effective: applying it to more than 600000 groups the minimum was determined in all but six cases. Finally, we employ the results of our calculations to compute the Lind-Lehmer constant for nearly 8 million additional p-groups. [ABSTRACT FROM AUTHOR]

We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 − 1 / ( 5.573412 log ⁡ | t | ) for | t | ≥ 2 . This represents the largest known zero-free region within the critical strip for 3.06 ⋅ 10 10 < | t | < exp ⁡ ( 10 151.5 ) . Our improvements result from determining some favorable trigonometric polynomials having particular properties, and from analyzing the error term in the method of Kadiri. We also improve an upper bound in a question of Landau regarding nonnegative trigonometric polynomials. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The case where $\alpha =0$ was investigated by Pólya; the case $\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\alpha }(x)$, where $0\leq \alpha \leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\alpha =1/2$is of particular interest. [ABSTRACT FROM AUTHOR]

P. Tur´an asked if there exists an absolute constant C such that for every polynomial ƒ ϵ Z[χ] there exists an irreducible polynomial g ϵ Z[χ] with deg(g) ≤ deg(ƒ) and L(ƒ - g) ≤ C, where L(·) denotes the sum of the absolute values of the coefficients. We show that C = 5 suffices for all integer polynomials of degree at most 40 by investigating analogous questions in Fp[χ] for small primes p. We also prove that a positive proportion of the polynomials in F2[χ] have distance at least 4 to an arbitrary irreducible polynomial. [ABSTRACT FROM AUTHOR]

Abstract: For a positive integer n that is not a power of 2, precisely the same family of convex polygons with n sides is optimal in three different geometric problems. These polygons have maximal perimeter relative to their diameter, maximal width relative to their diameter, and maximal width relative to their perimeter. We study the number of different convex n-gons that are extremal in these three isodiametric and isoperimetric problems. We first characterize the extremal set in terms of polynomials with coefficients by investigating certain Reuleaux polygons. We then analyze the number of dihedral compositions of an integer to derive a lower bound on by obtaining a precise count of the qualifying polygons that exhibit a certain periodic structure. In particular, we show that if p is the smallest odd prime divisor of n. Further, we obtain an exact formula for in some special cases, and show that if and only if or for some odd prime p. We also compute the precise value of for several integers by enumerating the sporadic polygons that occur in the extremal set. [Copyright &y& Elsevier]

Abstract: We show that the height of a nonzero algebraic number α that lies in an abelian extension of the rationals and is not a root of unity must satisfy . [Copyright &y& Elsevier]

The article discusses mathematical theories of designing the U.S. one-dollar coin. Several requirements of the coin design include polygonal shape, fixed diameter, maximal area and large perimeter. The isoperimetric problem applies in the determination of the polygonal shape with a perimeter covering inscriptions of maximum area, and which regular polygon is optimal with the proof of two formulas in geometry, the Heron's formula for triangles and generalized Brahmagupta's formula for quadrilaterals. The Reuleaux polygons concerns on the constant width for the coin. The isodiametric problems and the theory of Karl Reinhardt presents a geometric solution to the perimeter problem with odd number odd sides.

For a fixed integer n ≥ 2, we show that a permutation of the least residues mod p of the form f(x) = Axk mod p cannot map a residue class mod n to just one residue class mod n once p is sufficiently large, other than the maps f(x) = ±x mod p when n is even and f(x) = ±x or ±x(p+1)/2 mod p when n is odd. [ABSTRACT FROM AUTHOR]