From ‘there were three of us in this marriage’ to ‘recollections may vary’, a lot can be conveyed with a carefully worded phrase

As denials go, King Charles’s response to Earl Spencer’s allegations that Charles said after Diana’s death, “We’ll forget her soon enough”, was a masterclass in the diplomatic royal rebuke.

Rarely has Buckingham Palace responded to claims in countless books on royalty. But it was straight out of the block on this one.

Andrew Griffith is keen on clearing red tape. Good job he wasn’t part of a Treasury team that abolished the Office for Tax Simplification

On days like this, it’s not hard to be overwhelmed by existential futility. At the current rate of progress, the UK looks on course to wipe itself out far sooner than it would with any interference from AI. Everything feels hopeless. Years of minimal growth and a cost of living crisis. The tech bros and the bankers may be getting richer. The rest of us? Not so much.

And there’s no end in sight. Andy Burnham may offer better vibes, but he is fighting a losing battle against global forces – wars, energy prices, borrowing costs – that he can’t control. You can feel the hope drain out of him as each day passes. Mr Congeniality is not quite so congenial as he once was. All the economic forecasts suggest things are going to get even worse in the next year.

The genre-fluid singer answers your questions on being funky v lonesome, his infamous MTV interview with Thurston Moore and why he said yes to performing with a chain-smoking alcoholic robot

• Read Alexis Petridis’s 5* review of new album Ride Lonesome

You’ve been performing since the late 80s. What keeps you going? AmongstTheWaves
It’s a good question. I don’t really know any other life. I’ve spent a lot of my time trying to create a body of work, and out of that, maybe there’s a few songs at the end that are not bad. So I just keep going down into the mine, trying to come back with a couple of things worth keeping. It’s a lot of digging. If I had to stop and get a job painting houses or something, I would still make songs just for myself, because that’s how it started.

You tend to alternate between styles from album to album. What do you feel is closer to your intrinsic self, funky Beck or lonesome Beck? Mazzini
Probably lonesome. But the other side of things gets me out of my shell, stops me from being a little too ponderous, which I’m apt to do. The other side of what I do – Loser or Odelay – was more of a fluke. At the time that was what people liked. Now it’s kind of reversed – they prefer the more introspective, quiet songs, and stuff that’s a little more genreless. I don’t have any songs that they play at weddings, like Uptown Funk. I remember telling Mark Ronson, “Hey, you made a wedding song.” He was like, “Yeah, finally.” That’s a very elite category of song.

As concern about loneliness and social isolation grows, we spoke to people who have joined groups and set up activities

Amid growing concern about loneliness and social isolation, new research from the Institute for Public Policy Research found the number of young people in England with no close friends has risen fivefold since 2014, with one in 20 now reporting having none at all.

Making new friends as an adult is not always easy. We spoke to people who have taken deliberate steps to make new connections about what worked for them.

In early September, OpenAI announced it had solved a major mathematics problem that has stumped humans for nearly a century. The news left mathematicians reeling, and many expressed concern over what will be left for humans as AI becomes ever more adept at unravelling complex problems. Now 25 recipients of the Fields medal – often called the Nobel prize for maths – have signed an open letter expressing their fears of a ‘severe misalignment’ between AI companies and their field. To find out how AI is likely to upend maths – and how mathematicians might respond – Ian Sample speaks to Colva Roney-Dougal, professor of pure mathematics at the University of St Andrews

The manosphere says this type of suffering is targeted neglect, writes advice columnist Eleanor Gordon-Smith. But your suffering gives you the ability to make your part of the world better.

• Read more Leading questions

I’m approaching 30 and I’m still a kissless virgin.

I’ve suffered with depression and anxiety since childhood and my family isn’t the most functional or supportive. I don’t really have friends either, as I find it hard to connect with others due to being bullied and feeling that I’m not worthwhile or wanted. I was also homeschooled from 14 and moved to a different part of the country at 15 and a few more times since then, so I feel quite rootless and alienated.

Exclusive: PM, who is facing pressure over whether to approve North Sea oil and gasfields, makes strongest comments yet on issue

• British people understand the climate crisis threat. Today I’m empowering them to tackle it

• UK politics live – latest updates

The UK must fully commit to reaching net zero by 2050, Andy Burnham has argued in his strongest comments yet on the issue, saying it is no longer possible for Britons to believe they are not already affected by a changing climate.

Writing for the Guardian about new plans for a mass of community-led local green energy projects, the prime minister said the UK must “believe in itself”, and set out his determination to reach net zero targets, which have been rejected by the Conservatives and Reform UK.

Exclusive: Sisters of Nicholas Brandram recount 44-year-old’s deteriorating mental health in days before he took his life

Days before Nicholas Brandram, the man suspected of being the “Putney pusher”, killed himself, he told his family that the police investigation was “murdering him”.

He had “no fight left” in him, he told his sisters, Alexia Hicks and Sophie Voelcker, and could not see an end to the Metropolitan police investigation, or the online abuse.

In latest extracts from upcoming memoir, Diana’s brother suggests some in Buckingham Palace were ‘relieved’ at her death

The brother of the late Diana, Princess of Wales, has claimed in published extracts of his new memoir that King Charles sounded “giddily elated – like a lottery winner” in a phone call to him immediately after her death.

Earl Spencer also says he believed Buckingham Palace wanted William and Harry and himself to walk behind Diana’s coffin because the then Prince Charles was “adamant” he was going to walk and palace officials feared “members of the public might shout abuse at him”.

Mark Rutte said Britain will need to steadily increase defence spending to meet Nato target of 3.5% of GDP by 2035

Britain needs to show “a credible path” of annual increases in its defence budgets to meet a Nato target of spending nearly £30bn more on the military by the middle of the next decade, the head of the alliance said on Thursday.

Mark Rutte, Nato’s secretary general, said that Britain could not wait until just before 2035 to lift defence spending to 3.5% of GDP, in an unusually pointed intervention a day after he met prime minister Andy Burnham.