# Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

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### Number of integer solutions of a multivariate polynomial

Given a single-variable polynomial, we all know that the number of its roots is bounded in terms of its degree. A polynomial here is a polynomial with integer coeffecients, and a root of a polynomial ...
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### Littlewood's theorem on sign changes of $\text{li}-\pi(x)$

Is there any link to the paper " J. E. Littlewood, Sur la distribution des nombres premiers"? I cannot find anywhere on googling. What about the analogous to the theorem for small interval? ...
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### "3n + 1 problem" variant, using (n/2 + 3)

Background Variant "3n + 1 problem" Consider a mapping similar to the original "3n + 1 problem" (Collatz conjecture): $n → n/2 + 3$ for even n (variant rule due to +3) $n → 3n + 1$...
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### Can I define topology on Schwartz-Bruhat function space without using direct limit？

Given $F$ a Local field which is also a locally compact abelian group, we donote the Schwartz-Bruhat functions on it by $\mathcal{S}(F)$. Without using direct limit language and after simple ...
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### Show that $\sum_{n\le x}\max(n)=O(x)$

[An Introduction to Sieve Methods and Their Applications- M Ram Murty, pg.14, Q35,36] Let $\max(n)$ denote the largest exponent appearing in the unique factorization of $n$ into distinct prime powers....
1 vote
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### The divisor function and prime numbers

We introduce the divisor function $\sigma_1(n)$ which sums the divisors of an integer $n$ such that $n > 1$. The question is : what are the conditions required for $n$ satisfying the following ...
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### Sum of all integers up to $x$ with digit sum $t$

If $S(x,t)$ is the sum of all integers up to $x$ whose sum of digits is $t$, is there a way to calculate it? I mean for high arbitrary $x$. For example, $S(120,11) = 29+38+47+56+65+74+83+92+119 = 603$ ...
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### Dirichlet generating function of convolution that takes every k-th term

I need to find Dirichlet generating function $Z_k(s)$ defined below by Dirichlet convolution $f*g$, where arithmetic functions $f(n)$ and $g(n)$ are defined by Möbius function $\mu(n)$ and Von ...
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### A diophantine equation inspired in a conjecture due to Gica and Luca

In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$ over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and Mersenne exponents $x\geq 13$ such that $x^2-2$ is ...
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### Number Theory Undergraduate Thesis Topics Suggestions [closed]

I am currently looking for Number Theory undergraduate thesis topics. Do you have any suggestions or any websites/resources I could check? I am not a graduate student so I am looking for a relatively ...
1 vote
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### Weighted count of Egyptian fraction representations

This question emerged during an activity I ran for some middle school students this week; basically, it's about a way to "count" - with an appropriate kind of weight - the Egyptian fraction ...